No, they do not. However, there are many scooter rental services in the area, mostly for Disney and universal theme parks but i wouldn't be surprised if you were to call ahead and rent one from one of these companies they could bring you one to use while at old town.
Also, check with the facility/resort you are staying in. Ex. I was at Westgate and they had a contract with a company that allowed you to rent them and they were the ones that you could break down and take with you.
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