If you are flying, simply gate check your wheelchair. In other words, keep your chair with you, until you board. It will be tagged, either at check in or at the gate. The airline may even chose to ticket it at both. After you are seated, your chair will be taken to the baggage section of the plane and loaded. Very often you can see it being loaded. Since it is the last thing loaded, it will be the first item unloaded and will be at the door when you deplane. You likely will need it anyway for the large distances in the airport.
On days in WDW when you area feeling tired and need the chair (which very well could be every day; WDW is huge) keep the wheelchair with you until you are just ready to board the ride. There is always a place where it is stored with the other WCs. If it must be moved by cast members, it will be waiting for you at the exit. If your wheelchair blends in and looks like others, make sure to make the WC easily identifiable, usually with a blanket or such so it will not easily be mistaken and miss taken.
Our wheelchairs are our personal means for comfort a well as conveyance, being personally fitted for the user. A rental will do in a pinch but will never be matched for comfort.