Can the flight booking API book a trip where my customer flies into one city and comes home from another?

Yes, most flight booking APIs can build a trip where your customer flies out to one city and returns home from a different one. This is usually called a multi city or open jaw search rather than a return search, and you ask for it in a slightly different way.

A return search sends one origin and one destination with two dates, and the supplier works out the journey home for you. A multi city search lets you send each leg on its own, so you can say your customer flies out to one city, then travels home from another city on a later date. The supplier then prices the whole journey together, which is often better for your customer than buying two separate one way tickets, because airlines publish fares built for exactly this kind of trip.

What comes back does depend on the supplier behind the search. Some airlines and consolidators price these trips very well and give you one ticket for the whole journey, while others simply join one way fares together. Low cost airlines often have no through fare at all, so each leg ends up as its own booking with its own rules for changes and baggage. How many legs you can send in a single search can also be limited by the supplier, and a long, complicated routing usually takes longer to come back than a plain return.

If you sell this kind of trip often, try a few of your usual routes as a multi city search and look at whether the result is one fare with one ticket or separate tickets, and read the change and cancellation rules for each leg before you quote your customer.