Renting a double stroller is a must at Epcot if you want to extend your children's stamina. We've never done a full day with a 9am arrival, but we have started a day with a 10:30am Princess Breakfast at Akershus and stayed until after Illuminations. My children, 4 & 6, konked out in the stroller and we worked around that. Usually we would do a second sit-down meal, either a very late lunch at Teppanyaki, or a very early dinner at Le Cellier.
There is plenty to do at Epcot to engage your children. Besides some of what others have mentioned, there are performers at the various pavillioins throughout World Showcase. My kids gawked at the Chinese acrobats and enjoyed being chosen to wear a balloon sculpture made by a mime.
I tend not to like crowds, so we'll go against the grain as much as we can. When we arrive for a 10:30 bfast PS, we pass through Future World and pick up a fastpass for Test Track, which tends to be for later in the day. (BTW, my kids LOVE Test Track!) After our breakfast, we tour the attractions at world Showcase with very little crowds. (WS opens at 11am). As WS gets crowded, we then move to Future World and usually experience much shorter lines. Spaceship Earth is usually a walk on later in the day, but crowded in the morning. We end the night in WS because FW closes at 7pm.
I think the main reason for our success in getting the kids to last is that we don't force them to. Our attitude is if they last until Illuminations, great. But if they are tiring and we see it's not going to work, we have no problem with heading back early. I think it helps that our day is not scripted and that we approach it as an adventure.
Probably much more info than you needed, but that's how it works for us.

Good luck!