Hi Courtney! I can relate! My toddler never outgrew his picky eating, and now at age 14, he still only eats a handful of foods. Thankfully, Walt Disney World is very accommodating of finicky eaters.
On the Disney website, you can view the menus for the restaurants to determine if they offer items that will appeal to your toddler, and I've found that choosing restaurants with items that appeal to everyone in my family to be the best approach. Many restaurants are happy to prepare foods without sauces or certain ingredients upon request.
If a particular restaurant doesn't have your child's favorite foods on the menu, don't hesitate to ask for them anyway. My son loves
grilled cheese sandwiches, and he is able to order one at many Disney
restaurants, even when they aren't listed on the menu. For instance, 'Ohana brought him an
adult-sized portion of grilled cheese and french fries, and the Yak & Yeti
made him a grilled cheese on a hamburger bun. He was able to order a cheese pizza at the Hoop-Dee-Doo Revue and a grilled cheese at Mickey's Backyard Barbecue. Be Our Guest was able to make a grilled cheese for my son at the table-service dinner meal, but not at the counter-service lunch. The only other restaurants we've
encountered that weren't able to accommodate my son's request for his preferred foods are those in the Epcot World Showcase. The Rose & Crown made a grilled cheese sandwich for him, and Via Napoli happily serves him pizza without sauce, but restaurants like Teppan Edo and San Angel Inn, in our experience, aren't able to provide off-menu options. When my family dines at those restaurants, my son just waits while we eat, and then we buy him a meal from a counter-service restaurant.
Come visit us again here at the Moms Panel if you have more questions! We'd love to help!