First the procedure of the 12th Amendment, which became necessary when the electoral college was deadlocked in a tie between the Federalists and anti-Federalists
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, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate; The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted;–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; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice….
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, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President to the United States.
So let’s assume that Republicans implode and some states vote for Trump/Pence, some vote for Pence + New VP, some vote for New Pres / Pence and some refuse to vote at all. Those votes are locked in December 19th. Things could change before January 6th. Just to make things fun, let’s assume Hillary becomes unable to say the required Oath of Office or doesn’t have 270 votes.
How it is supposed to work is that on January 6th, the 51 ballots are announced to the new Congress that took office on January 3rd. They would vote (each state gets one vote – DC is a state for this vote) choosing among the top 3 (Hillary + 2 Republicans) until one person gets 26 states.
The “fail safe” against deadlock is that after the Presidential vote is announced, the same is done for VP – except the choices are limited to only the top two – making it impossible to have a deadlock, unless there is one. Unlike the House vote, the Senators just vote – but it requires 67 Senators for a quorum and 51 Senators voting for the winner. If Democrats do not show up, there will be no vote.
Perhaps anticipating these issues, the 20th Amendment layers on some more contingencies
Section 1. The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
Section 2. The Congress shall assemble at least once in every year, and such meeting shall begin at noon on the 3d day of January, unless they shall by law appoint a different day.
Section 3. If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.
Section 4. The Congress may by law provide for the case of the death of any of the persons from whom the House of Representatives may choose a President whenever the right of choice shall have devolved upon them, and for the case of the death of any of the persons from whom the Senate may choose a Vice President whenever the right of choice shall have devolved upon them.
There are several problems here. The biggest is that the procedure where the House or Senate can name a replacement for the Congressional selection only applies to the death of a candidate, not their resignation or disability.
The idea that if both President and VP remain undecided by January 20th, that Congress will resolve the problem by passing a law has a problem. There is no President to sign a law after noon on January 20th. The chance that Congress will be able to agree on an acting President is not good. Once an acting President is named (Barack Obama comes to mind), there is little motivation to resolve the deadlock.
The 25th amendment to handle the situation of a President being unable to be President (but not dead) only applies after the President is in office.
Maybe Russia will launch a nuclear tipped cruise missile at a US Aircraft carrier in the Mediterranean and make this discussion irrelevant.
We may find out in the next few months just how much it would take for the US Military to suspend the Constitution.