I don't want to imply anything to the OP about her child and his severe allergy, but I do want to be sure that the ADA is clarified. Something that would limit a major life activity is not something that occurs only when someone is in contact with an allergen. Someone's life activity of breathing is not substantially limited if he can breathe just fine without contacting the allergen. Breathing is considered as a substantial life impairment if someone cannot breathe on their own or has a breathing impairment severe enough that it limits mobility, self-care, working, etc all the time. The ADA is very limited on who is considered "disabled" under that law.
Since I have a fairly similar issue, I did some research a year or two ago. The DOJ's position (from reading settlements, mediations, legal briefs and speaking with someone from the DOJ) is that severe allergies can indeed be considered disabilities, subject to the same case by case screening that other disabilities are. As a result, allergen-related disability cases have gone both ways and there are dissenting options for almost all of the appeals. It's very much a case-by-case thing.
For example - you can't exclude a child from your daycare because they have life-threatening allergies and they must take reasonable precautions to ensure that the child avoids allergens. You can't require college students to pay for a meal plan that doesn't include food that is reasonably allergen free. You can't prevent people from carrying an epi-pen at a concert anymore than you could prevent them from carrying insulin.
Some Q & A from DOJ.
Whether a person has a disability is determined without regard to the availability of mitigating measures, such as reasonable modifications, auxiliary aids and services, services or devices of a personal nature, or medication. For example, a person with severe hearing loss is substantially limited in the major life activity of hearing, even though the loss may be improved through the use of a hearing aid. Likewise, persons with impairments, such as epilepsy or diabetes, that, if untreated, would substantially limit a major life activity, are still individuals with disabilities under the ADA, even if the debilitating consequences of the impairment are controlled by medication.
Now - I don't know that I would consider milk allergy cause to request a free fridge under the ADA (and it didn't occur to me to request one last time I was there) - but it sounds like Disney is happy to provide one for medical needs. And the more I think about it - the more it makes sense. When I stayed at Pop the only way to get food that was reasonably safe for me involved a half hour wait in the dining room. A fridge would have let me eat notably safer meals without an inconveniencing wait.
I'm not saying that I'm *entitled* to a fridge under the ADA. I'm saying that it's a nice thing to do that doesn't really cost Disney anything and makes life easier for people who have limitations. It encourages me to go back to Disney and stay on-property - the same way that a GAC would for people who might not meet ADA limitations but benefit from some consideration (a place out of the sun to wait for someone with photosensitivity for example).