Can I ask the airline for wheelchair help for my customer when I book through a flight booking API?

Yes, in most cases you can pass a request for wheelchair help or other special assistance to the airline through a flight booking API, but whether the airline acts on it is a separate matter from whether it was sent.

Airlines accept these requests using a set of standard codes, one for a passenger who needs a wheelchair to the aircraft door, another for one who cannot walk at all, and others for things like a passenger travelling with a guide dog or one who needs help because of poor sight or hearing. When your booking system supports this, your staff pick the right option while making the booking and the request goes onto the reservation along with the passenger name. Some requests, such as help for an unaccompanied child or carriage of medical equipment, are handled outside the normal booking flow because the airline wants to check them by hand first.

The honest limit is that a request is only a request. The airline usually accepts it as pending and confirms it later, and a few carriers, especially low cost ones, want the passenger or the agent to arrange assistance directly with them or at the airport instead. So the answer can also change from one airline to another within the same trip.

When you take a booking like this, send the request at the time of booking rather than afterwards, tell your customer it is pending until the airline confirms, and ask your supplier which of the assistance codes your content actually supports.