In the dead heat of August, my son, at age 4 even, had absolutely NO problem passing out in the stroller, and sleeping for prolonged periods of time. We often had to wake him up for rides.
That said, for a nice COOL nap, in MK, Hall of Presidents is great, as is the carousel of Progress. Once the sun goes down, the people mover is awesome. If the loud sound track doesn't keep him awake, a decent nap can be had in Philharmagic as well. Of course, none of those are particularly stroller friendly. Tom Sawyers Island could be though, as well as the small alley way off the right side of Main Street on your way into the park. Bonus, that is close to bathrooms, and food for the adults

Going towards Liberty Square from the Hub, on the right just after the bridge is a small seating area behind the building, near the pathway around the castle. There is a snack shop there (can't recall the name, it serves the fruit and nutella waffles though, maybe sleepy hollow something or other) that has decent shade and is stroller friendly. Nearly directly across from that, behind the Christmas shop is also a nice little area.
In EPCOT . . . Universe of Energy is a great air conditioned nap. So is Spaceship Earth, and living with the land

The "breezeways" between the Innoventions buildings often have nice cool air blowing through from the open doors, and shade if you can find an open bench there. The Mexico Pavilion (inside) is a nice place to relax for a bit. The seating area in the rock garden near the quick service in Japan is a favorite place of mine to relax. The American Adventure can be a wonderful place to nap as well. Finally, the rose gardens in the back of the UK pavilion.