Yes, in most cases you do. When a ticket is issued through a flight booking API, the airline booking reference, often called the PNR, comes back with the booking confirmation and is stored against that booking in your portal. It also appears on the ticket or itinerary you send to the passenger, so the traveller has it in hand along with the ticket number.
What the passenger can actually do with that reference depends on the airline rather than on us. Full service carriers usually let a traveller open their booking on the airline website or app using the reference and the surname, check in, print a boarding pass and sometimes change a seat. Several low cost carriers handle agency bookings differently and may only allow retrieval using their own reference, or may tie the booking to the contact details that were sent at the time of booking, so online check in does not always work the same way for them.
There are a couple of honest limits worth knowing. A small number of suppliers return their own reference first and the airline reference a little later, once the ticket is confirmed with the carrier. In those cases the booking in your portal is still valid, but the traveller may not be able to look it up on the airline site straight away. If a reference is missing or does not work, it can usually be traced back through the supplier.
Before you go live, place one test booking with each type of airline you plan to sell, then try retrieving it on that airline's own website to see exactly what your customers will experience, and ask us which of the airlines and suppliers available on your plan return airline references directly.