Do I have to refund their security deposit?

You are not obligated to refund the security deposit and the tenants can pursue collection of the security deposit from the prior owner in small claims court. Oftentimes it is easier to offer a cash-for-keys incentive to negotiate a move out date (typically faster than waiting the 90 days) and provide them with some funds to move since it may take time for them to get their deposit back from the prior owner.