If my customer wants to change their travel date after booking, can I do that through a flight booking API?

Yes in many cases, although a date change is more involved than making a fresh booking, and some tickets do not allow one at all.

When a change is possible, the system can look up the new flight, work out what the change costs and move the customer onto the later date, then give you an updated ticket. The cost is usually made of two parts, a change fee set by the airline and the difference between what your customer originally paid and what the new flight sells for on the day you make the change. That means a cheap seat which has since sold out can leave the new date costing more than the whole original trip. Both parts come from the airline rather than from you, and neither can be known for certain until the new flight is picked.

It gets less automatic with the cheapest fares, with tickets where part of the journey has already been flown, and with trips built from two different airlines, since those often have to be handled by a person speaking to the airline or the supplier instead of by the system on its own. Low cost airlines work differently again, as many of them treat a date change as their own process with their own charges. What your portal can do on its own therefore depends on the airline, on the fare bought, and on what your supplier passes through to you.

Before you promise a customer anything, look at the fare rules held against the booking, because they tell you whether a change is allowed and roughly what it will cost, and agree with your supplier which changes your own team can make and which ones have to go back to them.