One of my DDs used to have to have soya milk (we too now have Oatly instead as it's preferred for youngsters - another story!). When we stayed at the DLH they provided this at breakfast - you just had to ask for it from your waiter/ress.
Not sure if they provide it at other Disney hotels but it's worth an ask if you are staying at one of them. I think I emailed guest comms in advance to ask and they said while they don't put it out on view, they do have it at the DLH and just to ask for it.