Can I book a trip with more than two flights in it, like a customer flying to three different cities, as one booking through the API?

Yes, the flight booking API supports multi city trips, so a customer visiting three or four cities can usually travel on one booking instead of several separate ones. You send each leg with its own origin, destination and date, and the search comes back with fares that cover the whole journey. Keeping it as one booking is usually better for your customer, because the airline sees the trip as a single itinerary.

How many legs you can put in one search depends on the supplier behind the results. Most sources allow a handful of legs in a single request, and a few limit it to fewer than you might expect. Mixing airlines across the legs is often possible, but not always, and when it is not you may get back results that cover only part of the route. Low cost airlines in particular tend to price each leg on its own rather than as one journey.

It is worth knowing that a single multi city booking behaves as one unit later on. If your customer changes or cancels one leg, the airline rules for the whole ticket usually apply, not just the leg being changed. That can work in your favour on a proper connected itinerary, and against you when the legs were really separate trips that happen to be booked together.

The practical step is to run the exact route your customer wants as a multi city search and see what comes back, then compare it against booking the legs separately before you quote them.