It's surprisingly easy, as long as you're willing to give a little on things like soy/sugar/grass-fed/peanuts/dairy. If you aren't, then sit down restaurants, buffets, and eating things you bring in are your best bet. Lots of Paleo bloggers have done Disney. Google is your friend.
Table service is super easy as long as you're not choosing pizza and pasta. You can note that you'd like to speak with the chef if you're concerned about oils/soy/hidden starch, or, if you aren't so particular, you can just ask for extra veggies as a substitute for a starchy side and fruit as your dessert. I'm partial to Artists Point at the Wilderness Lodge and the buffalo steak. NOM.
Ohana is a great choice--similar to a Brazillian restaurant. Note on your ADR that you need a Grain Free/Soy Free meal and they'll roast your meat without the maranade. They may offer a substitution for the noodles/dessert, as well. Biergarten is another place that may work well for you. We're planning to try it on our next trip.
All the parks have quick service rotisserie chicken/rib places. You can also get Turkey Legs in each park. MK has a roasted pork shank at Gaston's Tavern that I'm REALLY excited to try.
Columbia Harbor House in MK has a salmon/broccoli/quinoa platter that may work for you depending on where you fall on the quinoa scale (you might be able to get extra broccoli instead if you ask.)
Sunshine Seasons at Epcot also has a really nice salad with beets and goat cheese (if you occasionally do cheese.) and a great pork chop dinner (not sure if you can swap out the cheddar mashed potatoes or get the chop ala carte). My other suggestion: SCHWARMA at Tangerine Cafe in Morocco. Love. It.
Liberty Square Market has an assortment of fruits, veggies, pickles (Crazy refreshing!) and baked potatoes--sometimes baked sweet potatoes.
For treats-- Main Street Confectionery usually has chocolate dipped fruits (strawberries, bananas, pineapple) and Big Top Treats near Dumbo has chocolate dipped frozen bananas with peanuts.
Bring some Uber Bars and jerky, maybe a PaleoKit or two, and you'll be GOLDEN. Have a great time!
(Search on the boards here for paleo, primal, and low carb. There are some other great ideas out there.)