I suppose animals could work in an alt-med type clinic, they accept almost anything else going so why ever not. I'd certainly not want to see one working in a proper hospital for a start how would it learn about dispensing, or drug regimes, or even delivering tea and coffee.
Maybe chimps could do the latter but then they might scare the patients. Not a good idea at all really. I'm sure you could teach a chimp to wash its hands on entering somewhere, but all that hair might be hard to check it's done right.