In addition to what other posters have said, Epcot has the most walking.
Also, it's not just the walking, it's the standing on concrete.
If you are out of shape, certainly, a healthy walk each day is a good idea. Last year, I was really good about hitting the Nautilus at least twice a week, and my legs never felt so good at WDW. My in-laws are always talking about all the swimming and sports they do. I'm not saying they brag or anything, it's just that they are much more into playing sports than I am. I was almost shocked that they couldn't come close to keeping up with me!
One lifesaver, for me, is to soak my legs at night. Either hit the hot tub or soak in the bathtub. You don't have to get your whole body wet, just your shins and feet.
Be certain to bring at least two good pairs of shoes! New, but broken in. For myself, I nearly always bring sneakers and quality sandals. I really like it when my feet can breath (instead of wearing socks all day). Maybe in January I wouldn't bother with sandals, but the rest of year, yes. I suggest something like Birkenstocks. Okay so maybe not that exact styling, but something of a similar design -i.e. with expandable straps that can be adjusted. Lots of other good shoe companies make good sandals. In any event, being able to trade shoes really helps. Even the best shoes need to be traded off periodically.
I also suggest buying your sneakers at a running store, where they will help you pick shoes that really fit, not the most expensive ones. Skip "The Finish Line" and "Foot Locker" type stores.