Not for president. We can have "special elections" for an empty seat in Congress, but it's never come up for the Executive Branch and I'm not aware of any mechanism that will allow for it.
If the President is out, then it's VP, followed by a long line of people in a defined chain of succession, last updated in 2006. It starts with VP -> Pelosi -> highest ranking Senator -> President's Cabinet (there's a specific order) and on and on. That person becomes acting President, not the actual President - that would come later, if at all. So, in a sense, an election would not be needed as the PotUS office would be automatically filled.
After that's settled then there can be discussion about either letting the acting President finish the term, or whether there can be some legislative or judicial procedure to allow an election.
Prior to inauguration I think the electors can choose another candidate. If it's after the electors submit their votes I think it either devolves to the House to decide among the the remaining three-highest-vote candidates but nothing like that has ever occurred here so I'm not entirely sure of that.