That's an excellent question, the answer to which, I think, is unclear. Article II of the Constitution specifies that only a "natural born citizen" is "eligible to" the "Office of President." On its face, this provision would not stop somebody who wasn't "eligible to" the "Office" -- for instance, somebody who was only 27 years old, say -- from running for President. But it could well be that each state (id perhaps not every state(?)) has its own requirements that only someone constitutionally "eligible to the Office" can be put on the ballot. As to that, I just don't know.
But, even there, what recourse, constitutionally, would there be if the Electors were to vote for, and elect, someone who was subsequently determined not to be "eligible"? My working assumption, and that's all it is, an assumption, is that the action of the Electors would be unreviewable. Said person, albeit otherwise "ineligible," would be the President. Impeachment and removal would be the only option to remove that person from office.
Good analysis. Have come to regard the Electoral College process, for good or ill, as the ultimate domestic sanitizing tool for liberal use by ineligible candidates or by candidates who successfully employed electoral fraud to put themselves in a position to take our highest office.