Can the flight booking API hold a seat for my customer while they make up their mind?

Yes, on some airlines, though not on all of them. The flight booking API can create a reservation without paying for it straight away, which most people call putting a booking on hold, and the airline then gives you a deadline by which the ticket has to be paid for and issued. If that deadline goes by with no payment, the airline cancels the reservation by itself and the seat goes back on sale.

The length of the hold is decided by the airline and not by your agency. It can be anything from a few hours to a couple of days, and it usually gets shorter the closer you are to the departure date. On very cheap fares, and during busy travel periods, there may be no hold offered at all. Most low cost airlines do not allow it either, because their fares are meant to be paid for at the moment of booking.

It is also worth knowing that a hold protects the seat but does not always protect the price. Taxes and airline surcharges can be worked out again when the ticket is finally issued, and if the fare your customer chose sells out in the meantime, the airline may ask for the difference. It is safer to treat a hold as breathing space for your customer to decide, rather than as a locked in price.

Check with your supplier which airlines offer a hold on your plan, and make sure your staff can see the payment deadline on every held booking so that nothing quietly cancels itself while they are waiting for the customer.