I think they can be reserved a year in advance. We have one reserved for our upcoming trip for just one day. It costs $44/day. You pick-up it up at the Bike Barn no earlier than 1:00p.m. and return it at 11:00a.m the next day. The Bike Barn phone number is 407-824-2742. You do have to reserve with a credit card and there is a 24hour cancellation notice policy.
As far as the trail to Wilderness Lodge, I am not sure. I have never been on it in a golf cart. Looking at the map it seems large enough that you would be able to.