If my customer wants to move their flight to a different date, can I do that through your flight booking API?

Yes, in many cases you can start a date change yourself, but whether it goes through depends on the airline and the fare your customer bought. A date change is not a simple edit to the booking. The airline has to undo the old journey and issue the ticket again for the new date, and its fare rules decide whether that is allowed at all, how close to departure it can be done, and whether a change fee and a fare difference apply. Cheap promotional fares are usually the most restricted, and some cannot be changed at all.

Where the airline supports changes through the connection, the flow is normally to search the new date, see what the change will cost, confirm it, and then receive an updated ticket. Where the airline does not support it that way, the request has to be handled offline with the airline, and you will see the booking updated once that is finished rather than straight away. Low cost carriers and full service airlines behave quite differently here, so there is no single rule that covers all of them.

One thing to watch is that the new date must have seats available in a fare the airline will let the ticket move into. If only a much higher fare is left, the difference your customer has to pay can be bigger than they expect, so always quote the full cost of the change before you confirm it.

Before you promise a date change to a customer, open the booking, read the fare rules attached to that ticket, and check what the software shows as changeable for that particular airline.