WDW does have sulfur water, which means that the source of the water (the well or aquifer) passes close to some sulfur deposits. My DH (a Ph.D. in chemistry) says that sulfur can be removed by either passing ozone over it or by adding bleach to the water. Judging by the taste of the water at WDW, I don't think they do either of those things. It could still be filtered, since only certain types of filters will remove sulfur. I have heard rumors that the water at WDW is treated with ammonia; that simply isn't true, since putting ammonia in water would give it a "fishy" taste and it isn't used for water treatment.
I don't know what chemicals your DD is sensitive to (and I'm not sure whether a person can be sensitive to sulfur); my kids and I all have sensitive skin and are fine drinking the tap water at WDW and also bathing in it. DH also drinks it although he complains about the taste. Water costs $2.50 per bottle in WDW, so if you're going to go the bottled route I'd suggest a trip to Wal-mart and picking up a case or two and bringing it with you.
Incidentally, sodas are made with tap water, and so are ice cubes. I think it's just about impossible to get away from the sulfur. If you're still concerned, you could contact the Florida Department of Environmental Protection.