This is pretty much just agreeing with what the other have posted.
There are no airlines in the US that allow someone to stay in their own wheelchair and have it strapped down. There are too many variations in wheelchairs to be able to make them safe for sitting in on an airplane.
There is a very narrow wheelchair called an aisle chair that a person who can't walk (or be carried) down the aisle can transfer (of be lifted) onto. The aisle chair is pushed to the person's seat and then they are transferred (lifted if needed) to the airplane seat. When he was alive, Christopher Reeve actually was a spokesperson for company that made aisle chairs, so if he could use one, most people with disabilities should be able to.
The person can stay in their own wheelchair until just before boarding. Depending on the size of the wheelchair, it may be possible for the person to bring their wheelchair into the front of the plane and then transefer. Many airplanes have a closet that a folded manual wheelchair will fit into. They are not required to have room for more than one wheelchair in the cabin and someone else may already be using it, so there are no guarantees.
All airlines have some sort of department that deals with special needs. Some have a link on their website that is easy to find; at others, you will need to do a search, but all have someone that handles special needs.