Hi there and welcome to YT . I am sorry your family is going through so much with this little puppy. Everybody is correct by letting you know that he is way too young to be away from his littermates and mother, let alone being sold in a petstore
My concern is that the reason why the pup is not eating is because he is sick, already... therefore, when he eats , vomits then is getting more dehydrated.. Parvo is deadly to young pups and hoping this is NOT what is happening to this little pup.
PLacing Kayro on his gums and offering him water at all times is good, and I would definately try and get him in to a emergency vet ASAP. the test for Pavo is very simple, then if he has it he can be treated right away, or you are going to have to make some other very tough choices about the pup.
It is common for a pup to have difficult reactions when placed into a new home , however this little one is way too young , and the voiting is what concerns me very much. He is not going to have a bowel movement if there has been no food in his belly to digest. He will also want to sleep because he is unwell. I wonder if he has a fever? that is why I feel like the best thing to do is to get him into a 24hr emergency vet , Just so they can rule out any major illnessess, and possibly have rehydrate the puppy with IV fluids.
He is so young and tiny that he does not have a lot to lose in his body, which is very dangerous.
Again, it is very important to keep the puppy warm too. Try to take him to an energency vet hospital.