Generally it is not. The law requires accommodation without proof. Most theme parks, including Disney, had previously required some form of Doctor's note indicating the need of accommodation (but not the medical diagnosis) as a way of originally getting around the ADA, until that too was challenged and removed.
I spent a couple minutes googling and it looks like the current IBCCES process is in fact being challenged as of January. We will have to wait and see where that goes.
Like I said though, accommodation cannot be denied for a paperwork technicality. Universal could get around that as long as they were still allowing people accommodation at the park, even without the IBCCES card, and just not advertising that fact. As long as no one had been harmed, there is no real basis for a suit right?
Otherwise, it would be discrimination to require a multi-day process in order to obtain accommodation. Unless maybe Universal and the other parks required ALL guest to apply for and receive the card prior to entry.