Like a BROKEN RECORD, I KEEP SAYING- Pence - could - have - resigned, before the presentation of the Electors to his position, rather than participate.
What happens next is up for debate.
NOTHING in the Constitution prevents this act.
Pence resigning would have made no difference. If there’s no VP on hand to preside, then, as Reily pointed out, the President Pro Tem of the Senate takes over.
This happened in 1969. VP Humphrey had probably expected to preside, but he was dispatched to Norway to represent the U.S. at the funeral of Trygve Lie. In his absence, the President Pro Tem (Senator Richard Russell) presided. Link to newspaper story: https://news.google.com/newspapers?id=BvsNAAAAIBAJ&sjid=x3sDAAAAIBAJ&pg=3226,4398264
Also, before the ratification of the Twenty-Fifth Amendment in 1967, there was no provision for filling a vacancy in the office of VP. Thus, there was no VP whenever the President or VP had died in office. This happened most recently on January 6, 1965.
Would a Pence resignation have “gummed up the works,” as freepersup suggests? No. It would have been purely routine – the President Pro Tem would have stepped in, as had happened in 1969 and several prior elections with no VP.