Can my customer change the travel date on a flight they have already booked through my portal?

Often yes, but not always, because it comes down to the fare rules the airline attached to that particular ticket. Many tickets allow a date change for a change fee plus any difference between the old fare and the price of the new date. The cheapest fares frequently allow no change at all, and on those the only real option is to leave the ticket unused or see what the airline offers as a credit, which is entirely the airline's decision and not ours.

Where a change is allowed and the supplier passes that ability through, your team can raise it from the booking itself, see the new date priced with the fee and the fare difference included, and take that extra amount from the customer before the new ticket is issued. On some airlines and with some suppliers the change cannot be made straight away and has to go to the airline or the supplier's own service desk, so what your portal does is log the request and show you the booking as pending until an answer comes back.

It is only fair to warn a customer that the rules vary by airline, by fare and sometimes by route, that low cost carriers handle changes in their own way, and that changes are normally only possible before the flight departs. Once a passenger has failed to travel, most tickets lose their value. Whether you see a change option on screen at all depends on which suppliers and airlines are enabled for your account.

Before you promise anything, open the booking and read the fare conditions shown against that ticket, and if you are unsure how a particular airline behaves, ask your support contact to confirm how changes are handled for it on your setup.