Can my customers choose their seat on the plane while booking a flight on my website?

Yes, in many cases they can, but how much of the seat map you see depends on the airline and on the flight content source sitting behind that search result. Seat selection is what the industry calls an extra service, and airlines decide for themselves how much of it they share through a flight booking API.

Where the airline does share a seat map, your website can show the cabin layout once the passenger names have been entered, and the customer can pick a window or aisle seat, see which ones are free and which carry a fee, and add it to the total they pay you. Low cost carriers almost always treat seats as a paid extra, so the charge shows as a separate line in the booking. Some airlines, and some cheaper fare types, do not release seat maps before ticketing at all. In those cases your customer books the flight normally and chooses a seat later on the airline website or at check in, and your staff can tell them that at the time of booking so there is no surprise.

It is worth being clear with your customers that a paid seat is confirmed by the airline, not by you, and that airlines do sometimes move passengers when the aircraft type changes. Seat fees are usually not refundable even when the ticket is, so that is a sentence worth putting in your own terms.

The next thing to do is tell us which airlines matter most for your routes, then run a few test bookings and see which ones return a seat map and which do not, so you know what to promise before you go live.