IF they are taking state or federal funds? Yes. If they are a private NPO ran by other outside groups such as churches? NO. The ADA in cases where government funding is involved should apply. But where it is not involved the ADA should not apply.
I presume they are taking federal funds, but I still don’t see how or why that should pertain. I’d personally do away with the excesses of the ADA and with federal funding for many, many things. But the idea that a social service agency can’t provide any government-funded services to anyone unless it provides the said services for everyone seems particularly baseless to me.