I generally am a low carb eater, have done Atkins, though currently in the "rest of your life" phase four of the hcg diet which allows me more fruit and the ability to throw in a few carby indulgences here and there more than Atkins ever allowed. But since I'm not sure how far you're able to go, I'll share my experience from when I was on Atkins in Disney...
Bun-less burgers you can do almost anywhere. If you have the ability to bring along "oopsie rolls" (google it if you're not familiar with it), you can then have your burger on your oopsie roll, which makes you feel more "normal." You can also do this with other types of sandwiches (as long as there's no sweet sauce like BBQ, and watch out for sandwiches on which melted cheese stick to the roll - those are frustrating!) A lot of places do fries or apples as a side, so obviously avoid the fries, but apples may be more doable, depending on your particular threshold.
Besides this, check out salads... Chicken salads were already mentioned. There were great make-your-own salads at ASMo that you could add tons of toppings to. I personally loved the taco salad (Pecos Bill's?) ... there's a topping bar there with lots of stuff you can top your salad with (or top your burger with, for that matter). I remember doing the Cosmic Ray's roast chicken (delicious!) with double green beans instead of a gb and a side of potatoes.
One good thing is that you do so much walking in Disney that you're likely to increase your carb count at least slightly without suffering. Consider sharing a "must-have" dessert here and there. Just don't go totally off track. Sf desserts may help keep the carb count lower too. The time I went on Atkins, over the course of a week, I had a few bites of potato, a couple sugar-free desserts, a bag of cheetos, and one really big ice cream shake that I thought would have completely killed my diet! When I got home, the scale was up just 1.5 pounds, and I lost all of that within four days of Atkins induction-style eating. Not too shabby!
Now that I've done the hcg diet though, I've actually had the experience of four straight days of disgustingly drastic carby binging, causing an 8-lb weight gain, and then doing four correction days to lose ALL 8 pounds! So I think for a short Disney trip, I could possibly just not worry about it - eat what I feel like, but just know I may be correcting it for a few days with strict guidelines when I return home.
So, do the best you can, but don't make yourself miserable! It is a vacation, after all! Just recognize you might have to be extra strict as soon as you return home.
Good luck to you! I admire you for wanting to stick to your diet in Disney. It shows commitment!