Yes, in many cases you can change the date rather than buy a new ticket, but whether it is possible, and what it costs, depends entirely on the fare rules of the ticket you bought and on the airline or supplier behind it. Cheaper promotional fares are often sold as non changeable, while more flexible fares allow a date change for a fee plus any difference in fare. The system follows those rules, it cannot override them.
Where a change is allowed, the usual flow is that you search for the new date, the system comes back with the fare difference and the airline change fee, and once you accept and pay, the ticket is reissued for the new date. Your customer normally keeps the same booking file, and a fresh ticket confirmation is generated so they have something to travel on.
Be aware that not every supplier supports date changes automatically. For some airlines, particularly low cost carriers and some content sources, the change has to be raised as a request and handled by a person, so it will not be instant and will depend on working hours on both sides. Changes very close to departure, or after the flight has already left, are often refused by the airline no matter how the request is made.
Before you promise anything to a customer, open the fare rules shown at the time of search and booking, and check with your account team which of your connected airlines allow changes directly in the system and which need a manual request.