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?
But who cares? Freedom of speech & religion are no big deal, as long as you trust those who would throw them away. Do you?