Can the flight booking API tell me if a flight time has changed after my customer has booked?

Yes, a flight booking API can normally tell you when an airline has changed the time of a flight your customer has already booked, but the way that news reaches you is not the same with every supplier.

The usual method is a retrieve booking call. You send the booking reference and the API sends back the reservation exactly as the airline holds it at that moment, including the current flight times. If the airline has moved a departure, swapped the aircraft or cancelled a leg, the new details appear there, and each flight carries a status that tells you the segment has been changed rather than simply confirmed. Many agencies run this call automatically each day for every booking that has not yet travelled and compare the result with what was stored at booking time, so a change cannot slip past.

Some suppliers go further and push the change to you instead of waiting to be asked. They place a message on a queue or send a notification to a web address you register with them, and your system picks it up shortly after the airline makes the change. That is the faster arrangement, but it depends on the supplier and on the agreement you hold with them, so please do not assume it is switched on until you have checked. Low cost carriers often behave differently here from full service airlines, and a handful pass changes on only to the contact left on the booking.

The sensible next step is to ask each of your suppliers which of these two methods they support, then agree how often bookings that have not yet departed should be checked and who contacts the customer once a change appears.