Disney is very mellow about the food rules, although they do enforce their rules against outside alcohol &/or glass containers quite strictly. You can bring somewhat larger coolers into Disney parks than the dimensions shown, but the catch is that you have to carry them; no wheelies allowed. I know several people who carry in backpack coolers such as this one because they are on special medical diets:
http://www.keepyourcooler.com/beachcooler.html
The size of that one technically is a 24-can, but WDW will allow it, as it's no larger than the usual daypack that many guests carry. However, once filled with enough food for a family of seven, I think you would find that such a bag would be hugely heavy to carry around all day.
The larger issue inside the parks is not carrying in your food, but finding a place to eat it. Using tables at counter-service restaurants is largely frowned upon unless you purchase at least something there, because you are depriving the paying customers of the table. At MK this is usually not as much of an issue because the restaurant at the bridge to Tomorrowland is usually not in operation, so those tables are often free at mid-day, but in the other parks it can be quite problematic. Benches are fine for snacking, but eating a full meal whilst sitting on a bench isn't the most relaxing experience. You can't do the blanket-on-the-grass thing inside the parks; the grass is roped off.
However you carry your food, don't do ice packs. (You're from the UK with a family of 7, so I assume that you plan to stay somewhere with a full kitchen?) Instead, freeze your drinks, so that they act as ice blocks for the other items. They will thaw sufficiently by the time you are ready to eat lunch.
Universal Parks are MUCH more strict about outside food. They enforce their rules exactly. In that case, better plan to picnic in the parking garage or eat at Citywalk.