If my customer wants to fly out with one airline and come back with a different one, can I sell that as one booking?

Yes, in most cases you can put an outbound flight from one airline together with a return from a different airline and sell it to your customer as one trip on your portal. The flight API can search each direction on its own and bring back the best options from every airline it is connected to, so your staff can choose one carrier going out and another coming back. To the customer it looks like a single booking with one total price and one confirmation.

What you should understand is that behind the scenes this is normally two separate tickets rather than one. Each ticket follows the rules of the airline that issued it. The baggage allowance can be different on each leg, the change and cancellation charges are worked out separately for each half, and if the outbound flight is delayed the second airline is under no obligation to carry your customer on a later flight. That is an airline rule, not something the software decides.

Whether you can do this at all depends on the supplier feeds sitting behind your portal and on the plan you are on. Some sources return mixed airline combinations on their own, others will only sell each airline as its own journey, and low cost carriers in particular are often sold as a separate ticket with their own payment and their own conditions.

Before you promise this to a customer, run a test search on your own portal for the route you sell most often and look at how the two halves are priced and what the fare rules say, then tell your customer plainly that the two legs are ticketed separately so nobody is caught out if a flight time moves.