Hi Mary! Bikes are not prohibited at the Disney resorts. In fact you are the norm at Fort Wilderness if you bring your bikes with you.
Please understand that biking at Walt Disney World needs parental supervision. There are so many out of state drivers, older drivers in the area, and large buses using the same roads. The best idea is to stick to the sidewalks and go slowly enough so you do not interfere with guests who are walking. Even in an area such as Fort Wilderness it is not safe to just let kids go out on their own.
Many of the moderate resorts will have bikes to rent so it might be best to look into that before you bring your own from home. You can rent a bike and/or surrey from the following resorts: Boardwalk, Caribbean Beach, Fort Wilderness, Polynesian, Old Key West, Port Orleans Riverside, Saratoga Springs, and the Yacht Club. These resorts have pathways or areas that are safer to ride then just roads. Please read more about bike rentals
HERE. Please also understand you will need to be responsible to keep your bikes safe while you are not using them. Resort rooms are not big enough to put them inside so locks are a must. It is always best to be safe then sorry no matter where you are.