As I've stated before, my wife and I are into fitness and very picky about nutrition, so we pack all of our own food, as well as that of our kids. Packing lunch, drinks and a bunch of snacks for a family of 5 just doesn't fit in a small brown paper bag. When our kids were younger, we used to hang it on the back of our double wide stroller (it made for one heavy stroller with two kids and a heavy lunch bag in it). Last year, we tried getting a cooler on wheels. We got in for a few days, until a CM stopped us at Epcot and said wheeled coolers weren't allowed because you can't have things that you drag behind you. Anything on wheels must be pushed in front of you, so you can see where it's going and not have someone trip over it. After thinking about it a minute, my wife and I agreed that made sense. So now we have a "gear stroller". It's a small folding umbrella stroller we got at Costco. We have a very large soft sided cooler bag (it's probably 24" long, 12" wide and 12" deep) that we pack all of our food in. We sit it in the seat and the handles of the cooler bag slide nicely over the stroller handles to "lock" the bag into place so it can't fall off the seat. We push that stroller with us just as anyone else would push a child in the stroller. Never once had an issue, and have received several surprised but positive comments from CMs and a few guests. More often than not, we'll park it somewhere where we know we'll be returning for lunch and walk around without it. For example, in Epcot, we park it at the Electric Umbrella and leave it there until lunch. We also will strap my wife's carry bag where we keep our ponchos, suntan lotion, autograph books, etc.. That way nobody is really carrying anything except for our phone and money. Works like a charm.
I will say that in our trip to WDW last month, we saw a number of people bringing in wheeled coolers that we had been previously told were not allowed. We thought about bringing ours back in, but agreed that the gear stroller is actually much more convenient than dragging something behind us.