Can my booking system handle it when my customer pays for a flight but the ticket never gets issued?

Yes, this can be handled properly, and it is one of the first things worth checking before you go live. A well set up flight booking flow keeps the payment and the ticket as two separate records, so if the money is taken and the airline does not return a ticket, the booking does not just vanish. It should sit in your system marked as pending or failed, with the payment attached to it, so you and your staff can see exactly what happened and act on it.

This situation is more common than most agents expect, and it is usually not anyone's mistake. The last seat on that fare can go while your customer is typing their card details, the airline can time out, or the fare can be withdrawn between the search and the booking request. Your payment gateway and the airline are two different systems, so one can succeed while the other does not. Good practice is for the system to alert your team straight away rather than leave your customer waiting with no email.

From there you usually have three options, and which one you can offer depends on the airline and the fare. You can try issuing again if the same fare is still open, you can rebook your customer on the next closest option and settle the difference, or you can refund. How quickly a refund reaches the card depends on your payment provider and the customer's bank, and if any money already reached the airline then the airline's own refund rules apply and it can take longer. Nobody can promise an instant return in every case.

Before you launch, ask to see how failed bookings appear to you, who gets notified, and agree with your team who contacts the customer and how fast.