Article II, Section 1, Clause 5:
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
or a Citizen of the United States, at the time of the Adoption of this Constitution
This was for the first round of Presidents. How they were conferred citizenship at the time of the adoption of the constitution I don't know.
For voters it was likely then as simple as it is today, just being present. All an immigrant has to do at the federal level today is sign a statement under penalty of perjury that they are a citizen. No proof is required.
With that, many (dem?) states claim if an immigrant has satisfied federal requirements "who are we to disagree".
Hell of a way to run a country. Yes, there have been suits in the past and pending even today.