Good idea to begin services with a new rescue who hasn't seen vet yet?

Hello, I have a potential client who contacted me yesterday in search of a sitter in anticipation/preparation for getting a rescue puppy in the next month or so. But she contacted me today - Saturday- informing me that she is on her way to visit/bring home a puppy way before she planned to. The owner wants me to begin services for the dog on Monday. I am torn whether this is a good idea or not and wanted to get some thoughts about the situation. My apprehensions center around a few things: 1) The puppy won't have had hardly any time to get settled in the new home or with the new parents and will most likely be frightened and confused. 2) There will be no idea of the temperament of the puppy, eg. aggression, etc. 3) Due to it being the weekend, the dog won't have had time to be checked out by a vet. 4) Depending on the age of the new baby, it may not be up to date on vaccinations, etc. I do not yet have my own insurance and am just covered, if at all, by Rover's services. What would any of you do in such a situation? I want to be able to help the new parents as this is quite an exciting but stressful time for them. But I don't know if I am putting myself in a vulnerable situation as little to nothing will be known or established with the pet. I want to do what is best for the little girl as well. It can be such a frightening and unsettling transition for the pup and the parents as well. Please let me know your thoughts. Thank you in advance!