We have rented a couple of times in early October and around Columbus Day and really didn't have any problems getting them at the parks. The only place we did have a potential problem was at the Magic Kingdom later in the afternoon. They were going to put us on a waiting list and then call us in the park when one became available, even though there was one sitting right there. Long story short the cast members working the desk didn't communicate w/ each other and the chair sitting there really was available for us so problem averted.
A tip for you, if you had a chair at the park early in the day and are going to your resort and then coming back to the same park, they will put your name on it and hold it for you. I don't know if this is an official policy, but the cast members did it for us twice at EPCOT. We didn't even ask, they inquired when we turned it in. That made our day!
We are in the same situation - my Mom doesn't need one at the resorts and trying to navigate it onto the bus would stress her out. This year we are staying at the Contemporary and plan to spend most of our time between the MK and EPCOT so we might rent from an outside company and wing it w/ a manual chair at the other parks. The monorail is easier to deal w/ than the bus. Good luck - PM me if you have any other questions.