It's unpopular, but DH and I wear boots (hiking/work) in the parks. Even wearing sneakers, my feet would be so sore at the end of the day, I'dhave a hard time getting going again if I sat, they just hurt so much. I'd also have muscle cramps in my shins/up the sides of my legs if we walked too fast. The boots have been a lifesaver, with more support around my ankles, a little heel, and more cushion. I also make sure I get potassium and hydration in the mornings before we get going (coconut water and low sodium V8 are both great for potassium), and hydrate throughout the day.
If your son is bigger than a stroller, consider a wheelchair... he could push it when he wanted to walk, and sit in if he needed that break. He wouldn't have to use it all the time. My grandmother did that her last trip.
DH and I also go out a few months ahead of Disney trips and try to start getting more walking in to gear up for it. In the summer, we'll go walk fairs and festivals... that gets us ready for heat and walking. We park at the very far end of parking lots when we go shopping... things like that.
And don't feel bad if you all just need to sit and take a break. Shows are great for that. As is people watching on a bench with some cold water to drink.