You do not have to be a guest of Ft. Wilderness to rent a cart.
I rented one a few weeks ago for 22 hours and was staying on the other side of Orlando for work.
Golf cart rentals start as early as 1pm and have to be returned by 11am at the end of the period you requested (the next day, 2 days, etc.). You are charged in full when you rent.
When I rented, the CM at the desk asked me my site number and I told them I was not staying at Ft. Wilderness. She wrote N/A on the form and didn't bat an eye. Didn't have to talk to a supervisor or anything.
Just drive over to Ft. Wilderness and park in the Hoop De Doo lot, then walk over to the Reception Outpost.
When you go in tell them you want to rent a cart. The best time to do this is around 1pm in the middle of the week. Weekends and holidays they are usually all rented out. (You can make a reservation if you know what day you want to rent). There will be paperwork to fill out and give them a credit card. The charge is currently $62 and change for 22 hours. Then you head outside with your paperwork and the folks there will show you how to use your cart and give you the key. Then go exploring to your hearts content around the Fort. If you are committed to coming back the next morning to finish out your rental, then at the end of the night just park your golf cart in the golf cart parking next to the car lot and get in your car and go back to your hotel. (Keep the key). The next morning, come back and ride around some more. Then right around 11am drive your cart down the return path, give them your key and make your way back to the parking lot.
If you are not committed to coming back the next morning to retrieve your cart, then that evening before leaving drive your cart down the return path, place your key in the after hours drop box and head on your way.