Hey there, Andy!
There are a number of options open to you for camping near
Walt Disney World. I checked in with my aunt and uncle, who are 5th wheel owners and like to spend winters as snow birds in the Sunshine State.
First, of course, is the absolutely incredible
Campsites at Disney's Fort Wilderness Resort. Starting at just $55 per night, you can't get a campground closer to the theme parks. The amenities are top notch, and the campsites are just beautiful.
Another nearby recommendation is the Orlando/Kissimmee KOA. With an onsite laundry, heated pool, fitness center, and other amenities, this park is about 20 minutes from the theme parks. It's clean and they offer discounts to members of the military, fire fighters, and police officers.
Kelly Park/Rock Springs is a little further away, as is Moss Park, which is equally as highly recommended. This is for shorter term camping, as you are limited to three consecutive weekends and no longer than 14 days. Steeped in wildlife and the Florida wilderness, you might see deer, bear, raccoons, and other beautiful creatures. Camping here is around $23 per night.
Your best bet to make sure a campground has what you are specifically looking for is to check out reviews from other campers and campground amenities on their websites. You'll find one that's practically perfect for your family.
Thanks for your question, Andy!
Anna