If my customer does not turn up for their flight, can I get any of the money back for them?

Usually yes, but only a part of it, and on some tickets nothing at all comes back.

When a passenger simply does not travel, the airline treats the ticket as a no show. On most tickets sold by the full service airlines the fare itself is lost, while the taxes and some of the charges collected on the airline behalf can be claimed back, because those are only due when somebody actually flies. The airline decides what falls into that second group, and it often keeps a handling charge out of whatever is returned. On the cheapest fares, and on many budget airlines, the whole amount is treated as gone and there is nothing to claim.

The claim is made against the airline rather than settled by your portal, so the money moves at the airline pace and not yours. Your portal raises the request, tracks it and shows you what has come back, but it cannot decide the outcome or speed it up. Waiting time varies a lot between airlines, and a claim raised long after the travel date is more likely to be refused, since airlines put a time limit on how late a no show refund can be asked for.

The most useful habit is to raise the request as soon as you know the passenger did not travel, and to read the refund line in the fare rules on the booking screen before you tell a customer what they are likely to get back.