Yes, no problem. I think the only limits are no glass and no outside alcohol. There is no alcohol at all at MK, other parks sell some in limited quantities.
I think the best strategy if you have your own stroller is to put the water/snacks in the bottom basket of your stroller. We like those large ziploc bags (2.5 gal). For the little ones, I also like those water bottles with 'sports' or squirt tops. It's really important to stay hydrated! The best way to do that is with water. Most water bottles can be refilled a few times, then tossed or recycled - which is better than sippy cups, because you don't have to worry so much about sanitizing them or losing them.
when you enter the parks, you will have to go through a bag search, so that's why the clear ziplocs work well. Sometimes you can fold your stroller withthe water bottles in the basket, if not, having all in one ziploc is pretty easy to deal with on the bus. If you've got jackets, have everyone wear theirs until after you get in to the park. Otherwise, just keep in mind that anything you carry in tothe parks- you'll have to carry all day! So it's wise to only bring the absolute essentials.
There's no shortage of places to get a wide variety of snacks at WDW either. Bottled water is expensive at WDW, so that's a good thing to bring with you...other snacks, in limited quantity, IMO. I'm not saying don't bring them, but just don't bring more than you think you'll eat.
Our family has a number of dietary limits too, but with a little planning, WDW is great. If you call them and talk about your needs they are VERY helpful! They serve Kosher, vegetarian, and can cover any food allergies - to name a few. The DIS has a listing of all the menus, but you can also call Disney if you need anything special. They just need like 48hrs. notice.
