He’d be a fool to of course with the national election coming and with the fact she’s Hillary (and as someone who donated to his campaign I’d be furious that any of the funds went to her), but hypothetically could he?
Yes.
If he chooses her as his VP candidate, their campaigns merge into the Obama-Clinton ticket, and they merge their assets/debits. Since he currently has a surplus and she has debts, some of his money goes to pay off her debts. (I think.)
If he doesn’t choose her as VP, he can still donate money to her campaign, though there may be a limit on the maximum amount he can contribute.
But he can always ask people to donate money to her campaign to pay off those debts. And given that he is the Democratic candidate, people/organizations may want to get on his good side by helping pay off these debts. In fact, donors who have already contributed the maximum to Obama but none to Clinton may now contribute more, giving to pay off the Clinton campaign debts. (This is true regardless of whether she is his VP or not.)