Can I show my customer a seat map and let them pick their seat when I book a flight through your API?

Yes, for many airlines you can, but how complete the seat map is depends on the airline and on the channel the fare comes through, so it will not be available on every flight you sell.

In practice it works like this. Once your customer has chosen a flight and you have entered the passenger names, your site can call for the seat map for each leg of the journey. Where the airline shares it, you get back the cabin layout showing which seats are free, which are already taken, and which ones carry a charge. Your customer picks a seat, and the choice is added to the booking along with any fee before the ticket is issued. Low cost carriers are usually good about this because seats are a big part of what they sell. Some full service carriers only release seat maps on certain booking channels, and flights operated by a partner airline under a codeshare often return nothing at all, in which case the sensible thing is to let the booking go through without a seat and sort it out with the airline afterwards.

It is also worth being clear with your customer that a seat is a request rather than a promise. A seat that looked free during the search can be gone by the time you confirm, and airlines move people around when they change aircraft type, so you can lose a seat you paid for. Paid seat charges are usually not refundable from the airline even when that happens.

The best next step is to try the seat map call in the test environment on the specific airlines and routes you sell most, and decide what your booking page should show when no map comes back.