A lot of people self-diagnose themselves with chlorine allergies, but such a thing is incredibly rare. While it is not impossible, I have had several people who claimed such an allergy swim in pools my company manages and had none of the side-effects they associate with chlorine. What typically causes discomfort and rashes are chloramines, which are the byproducts of chlorine reacting with contaminates. A strong chlorine smell indicates an abundance of chloramines and points to too little chlorine in a pool. It makes no difference what produced the chlorine, gas injection, solid or liquid forms of chlorine, or saltwater chlorine generators, once it is in the water chlorine is all the same. Pool water comfort is about properly balancing the water, assuring there is enough chlorine and the pH is properly controlled. There is also no such thing as a "saltwater filtration system", nor a pool that uses "no chemicals" that is open to the public in the US. All public pools MUST use an EPA approved sanitizer, either chlorine, bromine, or biguanide. Local codes dictate what those levels can be and they are regularly inspected and would be shut down if they are not kept within the guidelines. Cruise ships are a totally different situation than pools on land.
As was said above, most pools at WDW are bromine, though they can still be salt pools. The saltwater generator will produce chlorine which almost immediately reacts with the sodium bromide in the water to form bromine. Also the salt level in a saltwater pool in the US is 1/10th that of the ocean. They are about 2000-4000 PPM salt, most people can't even taste it. The ocean, on the other hand, is 35,000 ppm. So the experience is nothing like swimming in seawater.