Hi Scott!
Biking is certainly welcome at most of the Disney resorts. It is a fun activity for the family to enjoy together. The first thought is to make sure everyone is aware of the rules as there are many out of state guests at Disney who are unaware of the different routes and not always driving with their eyes on things going on around them. It is very important that everyone rides on sidewalks and not the roads.
When riding on the sidewalks please give pedestrians the right of way and alert them if you are coming up on them from behind. We bring our bikes to Fort Wilderness and know the dangers of bikes meeting people. At that resort we also have to be aware of walking pets!
Disney's Saratoga Springs Resort is one of the larger resorts that is spread out around a lot of territory. It is linked with sidewalks making bike riding possible and fun. Since you are traveling from my home state of Indiana bringing bikes will take a lot of space in your vehicle. I suggest you consider
Renting Bikes during your stay and this is possible at Saratoga Springs Resort. They can offer all size bikes including those with training wheels. Helmets and children seats are complimentary with a bike rental. Please also know that any child under 16 must wear a helmet according to Florida State law whether you are bringing your own or renting.
When renting bikes from a resort you will not be allowed to take them off property. With Saratoga Springs being so close to Disney Springs you might get permission to take them there. There were bike racks available at the entrance near Saratoga Springs but I have not been able to get confirmation if they are still there. Your resort recreation cast members will be glad to assist you with that information.
Enjoy the ride!