If my customer wants to cancel their flight, can I find out how much money they will get back before I actually cancel it?

Yes, in most cases you can ask for a refund quote through a flight booking API before you cancel anything, so you can tell your customer what they are likely to get back instead of guessing.

The way it works is that you send the booking reference to the refund quote request, and the reply comes back with the amount the airline is willing to return, the penalty it is holding, and the taxes that can be given back. Some airlines return the taxes even on a non refundable fare, and that is often the only money your customer will see. Anything you charged on top as your own service fee is your own decision and the airline knows nothing about it, so you have to add it back or keep it yourself.

Be honest with yourself that this does not work the same everywhere. Many full service airlines support a proper refund quote, while several low cost carriers do not offer one at all and will only tell you the outcome after the cancellation has already gone through. Even when a quote does come back, some suppliers mark it as an estimate, because the amount is only fixed when the airline processes the refund, and the money itself can take weeks to reach your account.

Before you promise anything, check with your supplier which airlines in your content support a refund quote and which ones do not, and get into the habit of showing the quoted amount to your customer in writing and having them agree to it before you send the cancellation.