No worries, sanddune ! It's perfectly possible to stay on Atkins while at Disney World.
I was there in January and had no problems. Mostly, I ate scrambled eggs for breakfast, sometimes with bacon, and then looked for a big salad for lunch. Columbia Harbor House has a fabulous salad with chicken, pecans, and pineapple (I picked out the pineapple). Or there is a chicken caesar salad at that big counter-service restaurant in Tomorrowland -- am I thinking of Cosmic Ray's? The food court thing at Epcot also has a huge garden salad. The Brown Derby at Disney/MGM has an incredible Cobb salad, and you can have them add chicken or shrimp. For dinner, I looked for meat and vegetables. My only issue was if I got hungry for a snack about mid-afternoonish. You could bring along several bars, for example, to keep away from the popcorn, which just smells far too good.
There are sugar-free desserts to be found if you look, but I was on Induction while I was there, and so I just didn't go there.
Lucky you!