If the hotel says it has no record of my customer when they arrive, who sorts that out?

The supplier you booked through sorts it out, not you and not the hotel, because the contract for that room sits between the supplier and the hotel and your booking sits with the supplier.

In practice you call or write to the supplier operations desk with the reference the API gave you back at the time of booking, and they contact the hotel and get the room reinstated or find another one nearby of a similar standard. Your customer is standing at a desk while this happens, so the thing that decides how it goes is not the technology, it is whether your supplier has a team answering at that hour. Every supplier will handle a case that comes in during office hours. Not every supplier answers at two in the morning in a country on the other side of the world.

This is worth understanding before it happens rather than during. Ask each supplier who you contact out of hours, how quickly they promise to answer, whether they will pay for a taxi and the difference in room rate if the customer has to be moved, and whether that promise is written into your agreement or is just goodwill. Keep the booking reference from the API and the voucher easy to find, because the first thing you will be asked for is the reference, and searching for it while a customer waits is time you do not have.

Before your first season, put the out of hours contact for each supplier somewhere your whole team can reach it, and check that the reference shown on the voucher your customer carries is the same one the supplier will recognise.