Virginia,
Thanks for asking your question to planDisney!
Yes, 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.
You will find that there are several of these water stations in each park, and they seem to add them as they design or renovate certain parts of the park. A few have been placed in locations where you really would have to be in line for a specific attraction to access the fountain, but you should still be able to find an available one in each park regardless. 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
EPCOT
Disney's Hollywood Studios
Disney's Animal Kingdom Theme Park
Remember, they are adding more of these regularly, so asking a cast member may help you in finding one nearby.
Write me back if you have any follow-up questions.
Have a Magical Vacation!
Doug