Yes she could. If the president dies, the vice-president becomes president. Article II.
So would Trump continue until a special election were held?
Unless something changes the current election results then Trump is out of office at noon on January 20th. There is nothing in the Constitution that allows for special elections for the president or for remaining in office once his term is over.
Biden wouldn’t be President if he keeled over in the next two weeks, and thus she would not succeed him. That is the dilemma in “what if” scenario that I was painting. In other words, The Presidency would not change because Biden was “President-elect”, and not President. She would not have a formal office, not even Senator. Talking about your Constitutional crisis. “There is nothing in the Constitution that allows for special elections for the president or for remaining in office once his term is over.” There is the conundrum: There has never been an instance where there was not a “President-elect” that would assume his duties on January 20th, and there is nothing in the Constitution that allows for that.