If my customer's card is charged but the booking does not go through, what happens to their money?

Yes, the money comes back, but it is not always instant and how quickly depends on your payment gateway and on the supplier.

A card payment and a booking are two separate events, so one can succeed while the other fails. When that happens your portal normally keeps the payment as a failed or unmatched transaction, with no ticket or voucher issued against it. That record is what proves the charge exists and it is what you use to chase the refund, so the first thing to do is find the transaction rather than assume nothing happened.

What happens next depends on where the booking broke. If it failed before anything was confirmed with the airline or the hotel, the amount can usually be voided or returned to the same card without the supplier being involved at all. If the supplier had already accepted the booking and then rejected it, or the fare changed at the last moment, the release has to come from the supplier side and that takes longer. Many gateways can be set to reverse an authorisation automatically when a booking does not complete, and that setting is worth turning on if yours offers it.

Timing is the part nobody can control. Even when the refund is released the same day, the customer's bank decides when it shows on their statement, and it is common for that to lag by several working days. Tell the customer this plainly at the time, because most complaints come from the silence rather than the wait. Ask your payment provider whether automatic reversal is switched on for your account, and get into the habit of checking your failed payments report at the end of each day so nothing sits there unnoticed.