Amber G. gave spot-on adice! You WILL have to have a written consent form signed by your client before their vet will treat the animal.
So, what I do on my initial meet-n-greet with a client is ask these things:
1) who is your vet? Before you accept the job make sure you know the address and phone number of the vet hospital. I always print out Google directions to their vet and have it handy. If some emergency pops up i do NOT want to waste time trying to find the vet. I have all the information ahead of time!
2) find a release form and have the client sign it. That way when you get to the vet you can hand it over so that they'll administer treatment.
3) I have a list of the three emergency/24 hour hospitals in town and know how to get to each one. I know which one is closest to my client. Ask your client at time of signing if, in case of emergency and the normal vet is closed, do they want you to take their pet to an emergency hospital (or, for instance, do they want you to call them first). Emergency visits are considerably more expensive, so have a plan AHEAD of time!