I'd have a conversation with the person and if we didn't agree on the requested methods I wouldn't care for the animal.
Im not going to take a animal that is not in the best of health and care for it. That opens you to many issues if the animal takes a turn for the worst in my care.
My vet requires a flu shot for all dogs being boarded. I don't get a flu shot and I sure never got my dogs one. However, when I needed to board them they told me it was required or my dog could not stay with them. I agreed and had it done so they got to stay. If I refused they would have refused me service and would have been well within their rights to do so.