We were in Disney late aug-early sep 2 years ago with an almost 2 year old. Next time we go when it's hot, I'll invest in a cooling towel. We didn't have one that trip, but did often dip a receiving blanket in the cups of free ice water from the quick service places and sponge it over him. We also drank tons of that free ice water. A couple of the days, we basically did one ride, got ice water, then sat in an a/c show, then got more ice water, then another ride, then more ice water, etc. We usually sent one person to get water while everyone else went to the bathroom/changed the toddler, found a cool place to sit, etc so getting the water never really cut into our time and it was so very needed anyway.
He wore sandals to the parks and took them off whenever he was in the stroller. He liked to throw them, so we started immediately taking them off and putting them in the basket so they wouldn't get lost. I'd recommend doing the same. Socks and shoes can get really hot on a little one and no one is going to care if a toddler is barefoot in their stroller. Make sure to apply sunscreen to feet though if they're going to be barefoot as in a stroller, as that's one of the body parts that will see the most sun!
You may want to consider taking advantage of the splash fountains in Epcot or the Casey Jr Splash area in MK. When it's really hot out the kids will dry pretty quickly in the sun, or you can bring some extra clothes for them to change into.
If your little one is sensitive about sun in their eyes (mine was/is), bring some sunglasses. Also, if your stroller canopy is adjustable, that can help to keep out some of the direct sunlight. We also used a thin receiving blanket to help block out the sun when he took naps. I wasn't able to find a good stroller fan that wouldn't eat through a million batteries a day, so we tried to find cool places to take the stroller while he was napping, while also using rider swap during this time.