If there is no vice president,
then there is no vote in the senate for the vice president.
Do you not remember when Agnew resigned,
and Nixon appointed Ford?
There would be no tie-breaker in such a case.
True, Nixon nominated Ford for VP. I am not sure what would have happened if either the House or Senate confirmation vote had been a tie. Maybe they would have to keep voting until either one side received a majority, or Nixon would have to nominate somebody else.