In most cases, if a child is physically fit, age 5-6 seems to be about the time the stroller can begin to be left behind. There are some other factors that swing it a bit earlier or later, such as your touring style and especially, weather; if it is really hot, a child may need it when he otherwise would not, especially for Epcot, which is a really large park with a lot of concrete. The converse is true, too; a kid who would otherwise not be able to manage might do better if the weather is unseasonably cool.
Also, when estimating a child's stamina in hot weather, take into account the climate you live in. If "really hot" where you live means temperatures in the 80's (F), or humidity that never goes over 40% or so, your child may need a stroller for a little bit longer than a kid who is used to a summer climate similar to that of Central Florida.
Children who are in poor physical condition may need to use a stroller up to about age 8, particularly in summer. Disney's rental strollers come in a range of sizes; you can rent one that will accomodate an avg. 12 yo, if need be. I'd say that most kids will use a stroller consistently up until just about the time they turn 5 at WDW; after that it may become a sporadic thing, it depends on the kid. Renting in a pinch will often work better than bringing your own at that point.
If you want to condition your child, especially for a summer trip, start going on walks with them (outdoors on pavement, not dirt or soft track surface), gradually working up the distance. If he/she can manage a non-stop mile without tiring, you should be OK. I've heard that the avg. visitor walks about 8 miles in a full 12-hour park day, but that is stop-and-start walking. I find that in summer, the kids I take can only really enjoy that on their feet if we take a long break in early afternoon; we need about 2 hours outside the parks to recharge. Otherwise tempers start to flare, and the fun quotient goes way down.
BTW, shoes *really* matter at WDW; choose wisely. Our whole family swears by good quality sport sandals, we found that they were much better than sneakers for theme park walking in warm weather. For cold-weather visits, hiking shoes work best for us.