If my customer cancels a flight, will the booking API tell me how much money they get back?

In most cases yes, though not always straight away. When a fare allows cancellation, a flight booking API can show you the airline's rules for that ticket, and for many airlines it can also work out what is due back to your customer once the airline's own charge has been taken off.

How dependable that amount is depends on the airline. Some airlines return a proper refund quote through the API, which you can show your customer before you cancel anything. Others only give you the fare conditions in words, and the exact amount is confirmed by the airline after the cancellation has been filed, which can take some time to come back and settle into your account.

It helps to remember that cancelling and refunding are two separate steps. Cancelling stops your customer travelling, the refund follows after that, and any service fee of your own is yours to decide and is not part of what the airline sends back. Even on a fare sold as non refundable, some taxes are often still returnable, so it is worth checking the rules rather than telling your customer that nothing is due.

Before you cancel anything, pull the fare rules for that booking through the API, and where the airline supports it ask for a refund quote first, so that the figure you give your customer matches what the airline is actually going to pay.