Yes, you can usually offer seat selection through a flight booking API, but how much you are able to do depends on the airline and on the supplier sitting behind the booking.
Where an airline shares its seat map, the API can return the cabin layout along with the seats that are still free, and you or your customer can pick one during the booking or after the ticket is issued. Some seats come at no extra charge and others, such as extra legroom or the rows near the front, carry a fee that is added to the booking as a separate service. That fee is shown to you before you confirm, so you can decide whether to pass it on and keep the total clear for your customer.
The limits are worth knowing before you advertise it. Not every airline makes seat maps available to agents, and a few only open seats at online check in, so the most you can do is record a preference such as aisle or window. Low cost carriers often charge for every seat. Some seats are always held back for crew, for families or for passengers who need assistance. On certain airlines a seat you choose is sent as a request and confirmed a little later rather than instantly, and if the aircraft is changed the seat can move.
Before you promise seat choice to a customer, check which of the airlines you sell most often support seat maps on your plan, and add a short line to your booking flow that tells customers when a seat is confirmed and when it is only a request.