As I was reading this answer, I realized that there's nothing in the Constitution to prevent a majority of Electors from voting for the same person for both President and Vice President.
The Twelfth Amendment says:
The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and of all persons voted for as Vice-President and of the number of votes for each...
The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed...
The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed...
The only restriction here is that one of the two people each elector votes for must not be from their own state. There is nothing to say that people from other states can't vote for the same person as both President and VP, and it would only take one such vote to enable it to be mathematically possible.
Given all this, what would actually happen if that were the way the vote played out?
There have been Presidents without a VP before, but as far as I'm aware, there's no provision for the President to not also be the VP. Assuming the candidate doesn't immediately resign as VP, would we have the President as tie-breaker vote in the Senate? Would the President be able to appoint anyone he chose as VP? Would there be grounds for the Supreme Court to get involved?