There’s nothing that says he has to pick one, or that he has to make his pick be part of his campaigning. But when the electoral college meets, they have to cast votes for the Vice President, and if nobody gets a majority of their votes (because the Republicans don’t have anyone to all agree on), then it goes to the Senate to pick one (which, in the current Senate, would presumably be a Democrat, which I imagine Trump wouldn’t want). One way or another, we’d end up with a Vice President on Inauguration Day.