I'm not sure how hard and fast this rule is - or, honestly, if it's still in effect - but Disney frowns on bringing food into the parks. And they do search your bags as you enter. I haven't known of anyone being refused entry because they had food, tho.
Having said that ... when our kiddos were much younger and we lived in the area, we'd routinely brings sandwiches, chips, fruit, etc., in the parks and buy drinks from vendors. We found some secluded spot where we could eat out of our bags.
But, there are no picnic facilities. You might consider tailgating ... returning to your car for lunch or whatever. WDW is so honkin' HUGE - the scope of the thing just doesn't lend itself to picnicing.
There are lockers available, though. Since this is your first visit, do LOTS and LOTS of research, or you'll find yourself standing around awestruck trying to figure out what to do. I'd recommend the Passporter Guides, or The Unofficial Guide to WDW, available at your friendly booksellers. And, of course, here on these boards, and I like
www.allearsnet.com, too.