Until 1967, there was no provision for filling a vacancy in the Vice Presidency. When Kennedy was assassinated and Johnson became president, he had no vice president - Speaker John McCormack was first in the line of succession. This was not a huge problem since the next presidential election was less than a year away. But Congress recognized this could be very problematic if a VP became president early in a term - as happened in 1945 when FDR died, leaving the presidency to Truman, who served the remainder of his term with no VP. This was less of a concern then when Vice Presidents didn’t do much but attend funerals and wait for the phone to ring. But given the expanded role and responsibilities of modern Vice Presidents, it’s important to fill the vacancy.
So, if Nancy Pelosi became President (oh, happy day!!!), she could indeed appoint a Vice President, with the advice and consent of both Houses of Congress.
And if the GOP Senate refused to approve Hillary, the Democratic House Caucus could elect her Speaker and have the same result. (I know, I know - but a girl can dream, can’t she?)
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