Hi Jonathan!
Generally speaking, you do not have to stay at a particular resort to use its
Children's Activity Center. However, some resorts do have additional activities that can only be enjoyed by guests staying at that resort. For example, the Grand Floridian typically offers "Adventure Time" which is an activity where children can create simple crafts and play games. However, the Grand Floridian also has the Mouseketeer Club which is open to everybody.
The Children's Activity Centers are available for guests ages 4 to 12. I highly recommend making an advance reservation. They can be made by calling 407-WDW-DINE. However, sometimes it is possible to just "walk in". My wife and i did that during our last stay at the Wilderness Lodge. When you call, be sure to ask about dinner. Depending on the hours you book, dinner may be included.
Using a Children's Activity Center is a great option for getting some time away from he children. Another option is Kid's Nite Out which provides in-room babysitting. To contact Kid's Nite out, please call (407/) 828-0920.
Have a magical vacation!