Those circumstances are provided for in the 25th Amendment to the Constitution, which was passed in 1967. Section 2 states:
"Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress."
And to re-iterate, a kamaltoeho-rizontal resignation will be taken as a sign of disgrace. Going to be lots of fireworks. Stock up on popcorn.