Can I change my customer's flight date after the ticket has been issued, or do they have to cancel and book again?

Yes, in most cases the date can be changed without cancelling, as long as the airline and the fare your customer bought allow it. This is usually called a reissue, and on your portal it sits against the existing booking rather than being a fresh search. You open the booking, choose the new date, and the system asks the airline what the new flight would cost.

What comes back depends on the fare. Cheaper fares often carry a change fee, and if the new date is dearer your customer pays the difference on top. Some of the very cheapest fares cannot be changed at all, and in that case cancelling and rebooking really is the only route. Those rules come from the airline, so the portal shows you what the airline says rather than a rule we invent.

Not every connection supports a change straight from the screen. Some airlines and consolidators still want the change handled by their own team, so your portal raises the request and the booking is updated once they confirm it. Low cost airlines vary the most here. Whether you can do it yourself in one go depends on which flight suppliers you are connected to and what your plan covers.

Before you promise a customer anything, make a test booking on your portal and look at the change option on it, then ask each of your flight suppliers which of their fares can be reissued online. Then you will know which requests you can settle yourself and which still need a call.