Do my customers get the airline's own booking reference so they can check the trip with the airline directly?

Yes, in most cases they do. A flight booked through your portal carries two references, the one your own system gives the booking so that you and your staff can find it, and the airline's own reference for the same trip. The airline reference is the one a passenger can type into the airline's website to see their seats, their baggage allowance or their check in.

Both normally appear on the confirmation the passenger receives and on the booking page inside your portal, along with the ticket number once the ticket has actually been issued. It is the ticket number, not the reference, that proves the seat is paid for, so it is worth teaching your staff to look for it before they tell a customer that everything is confirmed.

There are a few limits to be aware of. Not every airline works the same way, and some low cost carriers use a booking format that behaves differently on their own site. What the passenger is allowed to do with that reference is decided by the airline and by the fare you bought, so an airline may let them add a bag but refuse a date change because the ticket was bought through an agent. There is also a practical risk, which is that if a passenger changes something directly with the airline, your portal will not know about it and your records will fall out of step.

The sensible step is to check that your confirmation email shows both references clearly, and to tell your customers to come to you first before they change anything with the airline themselves.