Yeah... I know. Neither would Barack Obama (that pesky 22nd Amendment) but it's fun to think about. Meanwhile, I'm guessing it's going to be Harris. I'm pretty sure that the current environment almost requires that Biden choose a black woman or other woman of color.
I'm certainly no Constitutional law scholar, but the 22nd Amendment says:
"No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of President more than once."
It doesn't prohibit someone from being elected Vice-President, and it doesn't prohibit a Vice-President from assuming the office of President. It only prohibits being elected to the office more than twice, and distinguished between being elected to the Presidency versus holding the office or acting as President.
I'm sure someone has looked into that.