If a hotel is overbooked and turns my customer away at check in, who finds them another hotel?

Yes, your customer can be moved to another hotel, but the supplier who holds that booking is the one who arranges it, not your booking portal on its own. When a hotel cannot honour a room it has sold, it tells the supplier, and the supplier looks for a similar or better property nearby and confirms the change. Your job in that moment is to reach the supplier quickly with the booking reference and your customer's phone number, so the hotel and the supplier are talking to the same person about the same room.

What you can claim back afterwards, such as the taxi to the new hotel or the difference in the room rate, depends entirely on that supplier's terms and on the type of rate the room was booked on. Rooms bought through a wholesaler often have different rules from rooms bought on a direct contract, and a very cheap non refundable rate usually gives you less room to argue. No portal can force a hotel to hand over a room it does not have, so this part is always a supplier matter rather than a software one.

It helps to keep the booking record in your system honest while all this is happening. Save the new hotel name and the new confirmation against the same booking, and write a short note about what went wrong, so any of your staff can pick up the call later without asking the customer to explain it again.

Before your busy season, ask each of your hotel suppliers for their emergency contact and their written policy on relocation, and keep both somewhere your team can find them in a hurry.