Candace,
Thanks for asking your question to planDisney!
Yes, you can bring your own water bottle into the parks at Walt Disney World!
When it is time to refill your bottle, Walt Disney World has water bottle filling stations in each park and are adding more periodically. The old solution was asking a counter service restaurant for a glass of water, which they would gladly provide free of charge. But with the growth of water bottle usage, they clearly felt the need to add more self-serve options.
There are a few water stations in each park, and they seem to add more as they design or renovate certain parts of the park. The park maps do not currently reflect these locations, so you will want to ask a cast member where to find the nearest accessible fountain.
The resort hotels also are adding these as well, though I have not seen one in every resort, so I cannot promise anything. But staff at the resort's counter service restaurant can always help you out. By the way, the water is filtered, so that is a plus.
Of the ones I know to date, here are a few for each of the parks
Magic Kingdom Park: Inside
Cosmic Ray's Starlight CaféEPCOT: In the Odyssey Center
Disney's Hollywood Studios: In Star Wars: Galaxy's Edge, near the restrooms, and in front of
Slinky Dog DashDisney's Animal Kingdom Theme Park: In front of
Expedition Everest, and by the
Na'vi River JourneyWrite me back if you have any follow-up questions.
Have a Magical Vacation!
Doug