I'm not familar with
Disneyland but we've been to WDW several times and have some experience with sugar free items because my preschooler is a type 1 diabetic.
At least at WDW if you have a sit-down dinner reservation at one of their restaurants you can ask for foods to be prepared especially for your child without sugar.
I do want to note that many of the items we encounter are made with sugar alcohols (malitol/sorbitol) and that can give you TERRIBLE diarreha. I know there was a sugar-free brownie at WDW that tasted great but was made with these sweeteners and caused me a lot of problems (didn't bother my son because he ended up not wanting the brownie and I ate it!). We have no problems with aspartame (Equal) and Splenda however.