What happens to customer deposits or gift cards the failing business already took before I bought it?

Deposits and gift-card balances taken by the seller before the sale are generally the seller's, or the insolvent estate's, obligation, not something a buyer automatically inherits. In an asset purchase, the buyer typically only takes on the liabilities it expressly agrees to assume in the purchase agreement, so customers holding pre-sale deposits or gift cards would ordinarily need to make a claim against the seller or the estate rather than the new owner.

That said, some buyers choose to honour existing gift cards or deposits anyway, for goodwill and customer-relationship reasons, even without a legal obligation to do so. If that's part of your plan, it should be clearly reflected, and priced into the deal, in the purchase agreement rather than left as an informal expectation between you and the seller or receiver. Being clear about this upfront also helps manage customer expectations once you take over, since customers won't necessarily know the legal position on their own.