If I remember correctly, the House of Reps will vote on a President then on a vice president. It is technically possible to then have a Pres and VP who are of two different parties. Its happened a lot in our history in the first 100 years. But not in modern history.
It’s possible though.
Except in this case, it would only be the office of President that they are voting for. No matter what else occurs, Joe Biden is legitimately the Vice-President of the United States. He was elected to that position by the Electoral College on a separate, distinct, ballot from the Presidency.
The question is whether it falls to the House at all, or whether the position is simply vacant. If the election returns to the House, they have no choice, they must elect John McCain as President since he would be the only remaining qualified candidate (Article II, Section 1 specifies that if no candidate has a majority, the House will select from the five candidates receiving the most electoral votes; in this scenario, John McCain would be the only candidate remaining with any electoral votes).
However, since there really is no law or precedent for the situation, the most politically expedient solution is to simply vacate the office, not to re-adjudicate the election. In that case, Biden would succeed to the Presidency through the normal rules, leaving the Vice-Presidency vacant until a replacement is found and confirmed.