If I cancel a customer's flight, will the API tell me how much refund they are getting and when they will get it?

Yes, for most airlines the API can tell you the refund amount before you confirm the cancellation, but nobody can promise the exact day the money reaches your customer.

The normal flow is that you ask for a refund quote against the booking, and you get back what the airline will return once its own penalty is taken off, which taxes and fees are refundable, and anything your own portal has added on top. You show that to your customer, and only when they agree do you send the cancellation through. That way there are no surprises later and you have something in writing to refer back to if they query it.

Where it gets less tidy is who actually works out the refund. Some airlines answer automatically and the amount is final the moment you cancel. Others accept the cancellation straight away but calculate the refund by hand afterwards, so what you first see is an estimate that can move. Budget carriers often give a credit for future travel instead of money back, and some fares are not refundable at all, in which case only unused taxes may return. Then there is the payment itself. Card payments go back to the same card and the bank adds its own waiting period, while settlement through an account usually shows as a credit rather than cash. Which way each of your airlines behaves depends on the supplier and the fare rules, not on us.

Before you cancel anything, pull the refund quote and read the fare rules shown at search time, and ask your account contact which of the airlines on your portal refund automatically, so you can tell your customer honestly whether the amount is final or still an estimate.