How do you get a client to go to a doctor’s appointment?

Is it your responsibility to take your client to an appointment? Are you asking her to go and she is saying no. Just take her for a ride and end up at the doctors. If not your responsibility and you feel she needs to go tell your supervisor and have them call the contact on file.

take them to lunch afterwards as a reward if its a must go appointment and try to explain that.

If your client doesn't wish to go to a doctor appointment there is nothing you can do about that. Let her/his family know so that whomever is in charge of these things can notify the doctor of his/her patient's refusal to attend appointment. If there is an emergency care needed while you are attending your client, call 911.