We stayed offsite, and we found it easier to stay in the park and find something low key or relaxing to do during the kids’ normal nap/rest times- hall of presidents, spaceship earth, The Land and so on. It took about a half hour to get from the park to our car, get everyone packed back in (we have three in carseats) and then another 15 minutes to get back to the hotel, from which Epcot was visible. The good thing is, once you park your parking pass is good a any of the parks for the whole day. Which was nice, because when park hopping we found it so much easier to drive between the parks than use the park transportation. My son is obsessed with the Monorail and wanted to take it from Chef Mickey to Hollywood Studios, and we were basically told by the cast members that it would be a schedule suicide. We were glad that we drove over there, in the end. The process would not have taken as long if we weren’t coralling so many kids, so I guess it also depends on how old everyone in your party is, and also what time you get there/where you park in the lot. Our hotel shuttle was basically the same, and the first one didn’t leave until after rope drop, which was a no-go for us.
I would def. recommend a car if you can. I think it’s doable without one, but not optimal. We drove down there so we were stuck with ours