I have 5 kids- oldest is 9 and youngest is 2. They have been to Epcot 3 times and really enjoy the park. It's probably their next favorite after Magic Kingdom. They love all the big ticket attractions- Test Track, Frozen, etc. They also like to check out the shops and characters in the World Showcase. We usually only do 1/2 of the WS. However, we didn't have a great first visit there and I quickly figured out what to avoid with small kids. I am specifically commenting on Epcot in the heat, because we have only been there in 90+ degree weather. I think you need a different strategy when dealing with high temps and little ones.
On our first trip in May 2015, we ended up at Epcot on a very hot day during the Flower and Garden Festival. We arrived at the park around 11:15am (mistake #1), and we had made dinner reservations in Italy for 4:30 (mistake #2). We did Future World during the first couple hours, and that was fine. It was very hot and crowded everywhere, but manageable. Around 2pm we headed towards the World Showcase and it went downhill from there. All the character lines were very long and not in the shade. There was minimal walking space because of the festival. It was hard to push a stroller through (or 2!). We were so hot, and the kids were ready to leave by 2:45, saying they were hot and bored. Oh, did I mention I was also 6 months pregnant with #5 and my oldest was only 6 at the time?
We had to stick it out until that dinner res at 4:30 so we melted in the crowds until then, trying to pop into shops that had A/C and were not too crowded. We didn't have much success, especially since we had 2 strollers to park. We finally went into the restaurant when it opened and cooled off in the foyer. My 1 year old immediately dropped a full sippy cup of water that exploded all over the floor in the crowded foyer. She was just keeping things interesting for us, I guess. I think we ate at Tutto Italia (?). It was the only Italian place with an avail res at that time, but it was more of a fine dining feel. Not the kind of place you really want to be with a 6, 4, 3, and 1 year old. Glassware everywhere, a side plate for every side, etc. It would have been an amazing dinner for just my husband and I, but I could not get out of there fast enough with the kids. My 1 year old was in her glory grabbing at anything and everything. I think the servers sensed our stress level because we had our food very quickly and were in and out of that place in record time. The early 20's guy waiting on us was very sweet about it all. We high-tailed it out of Epcot after that and swore we wouldn't be back for years.
Well, the following July we scored a fastpass for the brand new Frozen ride. So, with four little girls 7 and under, that was that! We were going back. It was a totally different experience. We arrived at the park for rope drop and did Future World. Around lunch we went back to our resort to recharge and rest. We returned to the park around 4:30 or 5pm, refreshed enough to deal with the late afternoon heat before the sun starts setting and the nice summer evening settled in. Without a festival to celebrate, Epcot felt much more spacious. We had an amazing time. The kids loved Illuminations. We followed that same schedule the following year and it was great. The kids had a blast and we are going back for our 4th visit in a few weeks.
My tips to manage Epcot in the heat with little ones:
1. avoid festivals if you can (guessing some kids don't care about the flowers, exotic foods, and especially the booze LOL). If you are doing the fall one and it's not too hot, it's probably fine. Crowded and hot is a bad combo with little kids, especially if crowds are for something they aren't really interested in.
2. Get to the park at rope drop and get out of there for a few hours during the high afternoon sun. If you want to stay all day, maybe do the indoor stuff in future world while it's really hot.
3. Don't eat somewhere YOU can't relax. My kids ate leftover cold pizza in a little alley behind Morocco last summer and it was very relaxing for us. I am not saying you shouldn't do TS meals- that's a crazy suggestion. I am just saying it's important to go somewhere you can enjoy with your kids, instead of running interference the whole time. We didn't like the food at Biergarten, but the atmosphere was much better for the kids.
4. Grab a dessert treat and stay for the nighttime show. The heat is less overwhelming and it's really fun for the kids.