If you decide to rent at the parks just realize it will be more expensive. Make sure you get to the park at opening time then you should be able to secure a
scooter. Worse to worse if unavailable then you can rent a wheelchair but then you need to push it.
You could ask the offsite scooter rental company if you could drop off and pick up each day at their location but then you have to work with their hours of operation. Some offsite scooter rental companies don't have a store front; they may work out of a warehouse only and they may not allow it. There's time factors involved going back and forth to an offsite scooter company which may be very time consuming.
It's typical to just keep the scooter for the length of your stay. You do have to be responsible for charging the battery everyday. For most people it helps to have the scooter outside the parks, too.
If you need to maybe you could explain your situation a little better then we could offer specific advice for your situation.