I knew the counter service dining police would come into view with this post

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To the OP - yes, you can bring in a small soft sided cooler (size restrictions are posted somewhere) with sandwiches and non alcoholic beverages in non glass containers into Disney World. The only problem is that you might have to give a little bit of humor to the bag check attendant who asks if he/she can join you for lunch!
As for where to eat, I would advise you not eat inside a counter service restaurant when there are multiple families who have their purchased food from that establishment and are looking for a table. I think that's just obvious. Now, we have gotten a beer and a pretzel at Animal Kingdom from the kiosks just outside Flame Tree BBQ and then sat in the Flame Tree BBQ area to eat. However, there were plenty of tables when we did that and we have encountered some points of time where that area was mobbed so we moved on. Counter service police can also move on.
There are a lot of benches along the way at any Disney park. I wouldn't count on a full blown family picnic environment but there are places to go out of the way to have a decent time just for yourselves. And I think that's what the OP wants. But, the problem is, only the OP will know what kind of place will be acceptable. My advice - take your lunch, do your thing around the park and look out for what you want. And then do just that.