Is there a chance a hotel room my customer sees in the search is already sold out by the time they pay?

Yes, that can happen, although it is not common. The results you see in a hotel search come from the rates and availability your suppliers have given most recently, and availability keeps moving while your customer is still looking. In a busy season, or for a check in that is only a day or two away, a room can be taken by someone else between the moment your customer searches and the moment they decide to pay.

To stop this becoming a bad surprise, the portal checks the room again with the supplier just before the payment is taken. If the room is still there, the booking goes through at the price shown. If it has gone, your customer sees a clear message on the booking page and is sent back to the results to choose another room, and nothing is charged to the card.

It is rarer for a room to fall away after a booking is already confirmed, and what happens then depends on the supplier. Some suppliers send an update that appears against the booking in your panel, so your staff can see it and act on it. Others expect you to contact them directly. Either way you would then either move the customer to another hotel or ask for a refund under that supplier's rules.

If this matters a lot to your business, ask which of your suppliers return live availability at the time of booking and which work from a stored feed, because the two behave differently on stays that start the same day. It is also worth telling your staff to reconfirm any same night booking before they promise it to the customer.