Would he, as the running-mate of an ineligible candidate, chosen by that ineligible candidate? There is enough cans of worms to keep thousands of electric can openers running day and night for a long time. Every cabinet post, every ambassador, every presidential order... null and void.
He was chosen by the Democratic National Convention, which process is recognized by all 50 states in their rules for chusing Electors for Vice President of the United States.
And the concept of "running mate" has ZERO Constitutional significance.