We were there in 2008 and we asked the security guard about bringing in food and water. Absolutely okay, as long as you are not dragging a refrigerator and a hibachi. We were there in summer and froze the water (accidently), which felt great in the backpack. Take a couple of closed bottles and refill them at water fountains. There is a Walmart nearby, though I couldn't tell you how to get there. Great place to buy water and snacks, and I hear the souvenirs are cheaper there, too.
My daughter was five at the time and absolutely loved, loved, loved Epcot. But she is a nerd like her parents. Not interested in the characters, but we couldn't get her off Living with the Land. She even did the behind the scenes tour and loved, loved, loved it. Epcot also has "Soarin," "Test Track" and "Mission: Space" which are not to be missed and are worthy of a FastPass.
We didn't take a stroller, either. Her dad carried her on his shoulders when she got too tired. She, too, opened parks and closed them. She did fall asleep at Animal Kingdom on the way out (she ducked and never came back up), but that was our fifth day there and we had been on the go go go including a Space Shuttle Launch.
Consider sharing meals. For example, we go the DD chicken tenders, hubby and I got a Caesar salad. We put the chicken tenders she didn't eat into our salad and had a feast. Another time we ordered one taco meal; hubby and I split the tacos, then he made himself a taco salad at the fixings bar. Your kids could split a meal (unless they are big eaters), and you and hubby could share. Our downfall were the soft pretzels and the strawberry ice bars.
Find out if your hotel has a laundramat. That way you can take half the clothes, and get a 90 minute break while you wait for your laundry. Consider going to a park early in the morning,going back to the hotel for lunch, nap and/or swim, then returning to an open park after dinner for the fireworks and parade. I was there in 1984 and saw so many cranky parents dragging out overtired toddlers; my sister and I were as fresh as daisies because we took a siesta.
Another tricky thing to do: one parent stands in line for something like the Dumbo ride while the other parent does something else with the kids. As you get closer to the front, the kids can get in line with you. It's not really cutting, because the kids are riding with you. It is not as obvious with one child, though