I do know that the Constitution guarantees that each state maintain a republican form of government, but I'm not sure a Christian state would necessarily viotate that provision so long as non-christians retained the franchise.
The real problem would probably be in the details. Of what significance would it be to declare Vermont to be a Christian state? What would it mean? Any effort to close other churches would violate the 14th Amendment as would any effort to provide any special privileges to Christian citizens or their churches. Since no state benefits, privileges, laws or penalties could discriminate against non-christians, what would be the point of it?
ART. 22. Every person who shall be chosen a member of either house, or appointed to any office or place of trust, before taking his seat, or entering upon the execution of his office, shall take the following oath, or affirmation, if conscientiously scrupulous of taking an oath, to wit:
" I, A B. will bear true allegiance to the Delaware State, submit to its constitution and laws, and do no act wittingly whereby the freedom thereof may be prejudiced."
And also make and subscribe the following declaration, to wit:
" I, A B. do profess faith in God the Father, and in Jesus Christ His only Son, and in the Holy Ghost, one God, blessed for evermore; and I do acknowledge the holy scriptures of the Old and New Testament to be given by divine inspiration."
And all officers shall also take an oath of office.
The Constitution of the State of New Hampshire 1776; 1784, [1792] Stated:
ART. III. When men enter into a State of society they surrender up some of their natural rights to that society, in order to ensure the protection of others; and without such an equivalent the surrender is void. (This flys in the face of Libertarian Claims.)
ART. V. Every individual has a natural and unalienable right to worship GOD according to the dictates of his own conscience and reason; and no person shall be hurt, molested, or restrained in is person, liberty, or estate for worshipping God in the manner most agreeable to the dictates of his own conscience, or for his religious profession, sentiments, or persuasion; provided he doth not disturb the public peace or disturb others in their religious worship. (Notice freedom was granted to Worship GOD according to their own conscience, Atheism would not be protected under this article.)
ART. VI. Morality and piety, rightly grounded on evangelical principles, will give the best and greatest security to government, and will lay in the hearts of men the strongest obligations to due subjection; and as a knowledge of these is most likely to be propagated through a society by the institution of the public worship of the DEITY, and of public instruction in morality and religion; therefore, to promote those important purposes the people of this State have a right to empower, and do hereby fully empower, the legislature to authorize, from time to time, the several towns, parishes, bodies-corporate, or religious societies within this State, to make adequate provisions, at their own expense, for the support and maintenance of public protestant teachers of piety, religion, and morality: (Sounds like the State Sanctioning of religion to me.)
PART II-- THE FORM OF GOVERNMENT
SENATE
Provided, nevertheless, That no person shall be capable of being elected a senator who is not of the Protestant religion, and seized of a freehold estate in his own rights of the value of two hundred pounds, lying within the State, who is not of the age of thirty years, and who shall not have been an inhabitant of the State for seven years immediately preceding his election, and at the time thereof he shall be an inhabitant of the district for which he shall be chosen.
HOUSE OF REPRESENTATIVES
. . . Every member of the house of representatives shall be chosen by ballot, and for two years at least next preceding his election shall have been an inhabitant of this State, shall have an estate within the district which he may be chosen to represent, of the value of one hundred pounds, one-half of which to be a freehold, whereof he is seized in his own right; shall be at the time of his election an inhabitant of the town, parish, or place he may be chosen to represent; shall be of the Protestant religion, and shall cease to represent such town, parish, or place immediately on his ceasing to be qualified as aforesaid.
EXECUTIVE POWER-- PRESIDENT
The President shall be chosen annually; and no person shall be eligible to this office, unless at the time of his election, he shall have been an inhabitant of this state for seven years next preceding, and unless he shall be of the age of thirty years and unless he shall at the same time have an estate of the value of five hundred pounds,, one-half of which shall consist of a freehold in his own right within this State, and unless he shall be of the protestant religion.
Constitution of the State of North Carolina (1776)
XXXII. That no person, who shall deny the being of God or the truth of the Protestant religion, or the divine authority either of the Old or New Testaments, or who shall hold religious principles incompatible with the freedom and safety of the State, shall be capable of holding any office or place of trust or profit in the civil department within this State.
The Fundament Constitutions of the Carolinas (1663) Which preceded Both Seperate Carolinas stated: No man shall be permitted to be a freeman of Carolina, or to have any estate or habitation within it that doth not acknowledge a God, and that God is publically and solemnly to be worshiped.
The Constitution of the State of South Carolina (1778) Stated:
XXXVIII. That all persons and religious societies who acknowledge that there is one God, and a future state of rewards and punishments, and that God is publicly to be worshipped, shall be freely tolerated. The Christian Protestant religion shall be deemed, and is hereby constituted and declared to be, the established religion of this State. That all denominations of Christian Protestants in this State, demeaning themselves peaceably and faithfully, shall enjoy equal religious and civil privileges. To accomplish this desirable purpose without injury to the religious property of those societies of Christians which are by law already incorporated for the purpose of religious worship, and to put it fully into the power of every other society of Christian Protestants, either already formed or hereafter to be formed, to obtain the like incorporation, it is hereby constituted, appointed, and declared that the respective societies of the Church of England that are already formed in this State for the purpose of religious worship shall still continue incorporate and hold the religious property now in their possession. And that whenever fifteen or more male persons, not under twenty-one years of age, professing the Christian Protestant religion, and agreeing to unite themselves In a society for the purposes of religious worship, they shall, (on complying with the terms hereinafter mentioned,) be, and be constituted a church, and be esteemed and regarded in law as of the established religion of the State, and on a petition to the legislature shall be entitled to be incorporated and to enjoy equal privileges. That every society of Christians so formed shall give themselves a name or denomination by which they shall be called and known in law, and all that associate with them for the purposes of worship shall be esteemed as belonging to the society so called. But that previous to the establishment and incorporation of the respective societies of every denomination as aforesaid, and in order to entitle them thereto, each society so petitioning shall have agreed to and subscribed in a book the following five articles, without which no agreement fir union of men upon presence of religion shall entitle them to be incorporated and esteemed as a church of the established religion of this State:
1st. That there is one eternal God, and a future state of rewards and punishments.
2d. That God is publicly to be worshipped.
3d. That the Christian religion is the true religion
4th. That the holy scriptures of the Old and New Testaments are of divine inspiration, and are the rule of faith and practice.
5th. That it is lawful and the duty of every man being thereunto called by those that govern, to bear witness to the truth.
And that every inhabitant of this State, when called to make an appeal to God as a witness to truth, shall be permitted to do it in that way which is most agreeable to the dictates of his own conscience. And that the people of this State may forever enjoy the right of electing their own pastors or clergy, and at the same time that the State may have sufficient security for the due discharge of the pastoral office, by those who shall be admitted to be clergymen, no person shall officiate as minister of any established church who shall not have been chosen by a majority of the society to which he shall minister, or by persons appointed by the said majority, to choose and procure a minister for them; nor until the minister so chosen and appointed shall have made and subscribed to the following declaration, over and above the aforesaid five articles, viz: "That he is determined by God's grace out of the holy scriptures, to instruct the people committed to his charge, and to teach nothing as required of necessity to eternal salvation but that which he shall be persuaded may be concluded and proved from the scripture; that he will use both public and private admonitions, as well to the sick as to the whole within his cure, as need shall require and occasion shall be given, and that he will be diligent in prayers, and in reading of the same; that he will be diligent to frame and fashion his own self and his family according to the doctrine of Christ, and to make both himself and them, as much as in him lieth, wholesome examples and patterns to the flock of Christ; that he will maintain and set forwards, as much as he can, quietness, peace, and love among all people, and especially among those that are or shall be committed to lids charge. No person shall disturb or molest any religious assembly; nor shall use any reproachful, reviling, or abusive language against any church, that being the certain way of disturbing the peace, and of hindering the conversion of any to the truth, by engaging them in quarrels and animosities, to the hatred of the professors, and that profession which otherwise they might be brought to assent to. To person whatsoever shall speak anything in their religious assembly irreverently or seditiously of the government of this State. No person shall, by law, be obliged to pay towards the maintenance and support of a religious worship that he does not freely join in, or has not voluntarily engaged to support. But the churches, chapels, parsonages, globes, and all other property now belonging to any societies of the Church of England, or any other religious societies, shall remain and be secured to them forever. The poor shall be supported, and elections managed in the accustomed manner, until laws shall be provided to adjust those matters in the most equitable way.
Constitution of the State of Pennsylvania (1776)
SECT. 10. A quorum of the house of representatives shall consist of two-thirds of the whole number of members elected; and having met and chosen their speaker, shall each of them before they proceed to business take and subscribe, as well the oath or affirmation of fidelity and allegiance hereinafter directed, as the following oath or affirmation, viz:
I do swear (or affirm) that as a member of this assembly, I will not propose or assent to any bill, vote, or resolution, which stall appear to free injurious to the people; nor do or consent to any act or thing whatever, that shall have a tendency to lessen or abridge their rights and privileges, as declared in the constitution of this state; but will in all things conduct myself as a faithful honest representative and guardian of the people, according to the best of only judgment and abilities.
And each member, before he takes his seat, shall make and subscribe the following declaration, viz:
I do believe in one God, the creator and governor of the universe, the rewarder of the good and the punisher of the wicked. And I do acknowledge the Scriptures of the Old and New Testament to be given by Divine inspiration.
Several Other State Constitutions did have established religion. Not a specific denomination mind you, but Protestant Christianity.
It's unlikely that they could do so today. The Supreme Court, which outlawed school prayer on the basis of the 14th Amendment, certainly would not stand for established churches.
The Supreme Court decision is here and here. I have my doubts about it, but it is something one should be familiar with if one discusses the topic.
This was Jefferson's great mistake, if one assumes that he was a great supporter of state's rights. He and Madison campaigned against established churches, and disestablished Virginia's Anglican/Episcopalian Church. They needled the New England States about doing away with their Congregationalist establishments.
Had Jefferson and Madison really wanted stronger state governments and a weak central government, they would have encouraged states to retain their established churches. What would keep states from coalescing into one uniform and homogenous nation more than separate state churches? Sectarian differences would have hindered New England, the Middle Atlantic and Middle Western States or the Southeast and South Central states from forming larger sections arrayed against each other.
To say that states could still establish churches after the 1830s is to miss the point. A right unexercised is a right lost. If you disestablish your state church you won't be able to get it back. And the bulwark against amalgamation that state religious establishments provided is also lost.
If the European experience is any guide, the burden of supporting established churches would have made us a less religious people, which would also have pleased Jefferson. In part, Jefferson didn't understand what would further his own goals. But also, he was more of a centralizer than some give him credit for being, although in a backhanded way. During the 1790's Jefferson wanted to use the states as a bulwark against Federalist power in the central government. Later, when his own party was in power, he was less concerned with this problem. But beyond purely constitutional questions, Jefferson's idea of freedom would sweep away local cultures, elites, and restrictions and help to create -- for better or for worse -- a freer, but also more homogenous, more unified nation. Religious disestablishment was an important part of that idea. The moral is that individual freedom and local autonomy aren't always in the same camp.
Thomas Jefferson got a letter from a citizen in Pennsylvania who was very upset that somebody said that the US constitution prevented Pennsylvania from naming the quacker religion as being the state religion. Jefferson wrote the man and said that the US constitution did not restrict the state legislatures from doing that. He said Pennsylvania was free to do that precisely because of 'the wall between church and state'. It was the federal government that was restricted from interfereing. That's what the term 'wall between church and state' actually meant when Jefferson first wrote it. That letter is the original source of the phrase. Today it is perverted to have a different meaning in many people's eyes.
The significance of these things is that any state in America is free to set its' own character through its' own laws. For example, in the state of california they've decided that in schools when little boys and girls are 11 or 12 years old that an adult homosexual would come to class as a positive role model and to get young boys thinking if they too might be gay like that nice man. In other states, like Kansas, they may choose to teach different things in the schools.
Obviously the federal government has become too much of a monster to tolerate the original meaning of the constitution.
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In my opinion, anyway. 
You don't hear a whisper here in Texas from our Rino governor about the borders and their control.
Better read up on your history. I'm not saying that Christianity is inherently tyrannical, but history does show that there have been very tyrannical governments claiming to be basing their legitimacy on "Christian" principles. I have no desire to see state religions. Politics unfortunately corrupts Christianity.
This has little relevance today except to point up the outrageous, glaring, and oft-amusing hypocrisy of some groups--particuarly atheist libertarians--who love to boast of their reverence for the wisdom of the founding fathers but insist that only a group of bigoted idiots would allow a state to establish and maintain a religion.
However, just because they had the legal power to do it, it does not mean that they should do it. My reading of Locke(who supported toleration, rather than religious liberty), Jefferson(who was not always consistent), Paine(a consistent apostle of freedom of conscience), Mill(preacher of the errors of infalibility), and others have convinced me that any maintained establishment of religion or religious tests for office, not only violates the individual's liberties, but undermines the general happiness as well.