Constitutional FAQ Answer #72
Q72. "What is the process for replacing the Vice President if the President leaves office permanently, and not temporarily as with an illness, etc.?"
A. The President appoints a new Vice President on vacancy of that office, as directed in the 25th Amendment. So if Bush were to resign, Cheney would become President, and then appoint a VP. If Cheney were to then resign, that VP would become President and appoint a VP. If Cheney resigns, Bush appoints a VP. And so on.