Good evening, Amy.
Immunity for the puppies will depend on what the mother received and how her immune system is, as well. If she has not been recently vaccinated by a veterinarian, she may not have great immunity against parvo, distemper and so forth. Studies do show that many dogs keep immunity as time goes on, but immunity is often boosted with repeat vaccines. If you're interested in receiving those answers on her level of immunity, you can certainly have titers performed to determine the levels of immunity for mom. If she does have reasonable levels, there's a good chance she's passed immunity on to her puppies, too.
That being said, 4 week old babies are a little young for parvo, distemper and some of the other afflictions that we see puppies acquire. We generally see this around the 8-10 week mark when puppies are being weaned and their immune system beings functioning for itself. As well, if you don't have a known exposure for these afflictions, the risk is even lower.
The vaccine portion of your question being discussed, do you currently have the mother on any dewormer? Have the puppies been dewormed yet? If so, what is mom on, how long has she been on it, how often are you giving it and how much? And the puppies?
If the puppies, themselves, have not been dewormed, this might give you an answer for why you have a vomiting puppy. Intestinal parasites are easy to pick up from mom and also from the environment. In some cases, there can be such a heavy parasite load in the environment that even while the mother is on a deworming regimen, there may still be adult worms present in her body and the puppies can acquire them. A recent fecal examination having been performed, and being negative, would be the most useful diagnostic here (please note: you do not have to see worms in the stool of mom or puppies for them to be present, we check for microscopic eggs during a fecal exam which cannot be seen with the naked eye). On top of this, puppies can always eat too much or have congenital issues that begin to cause them issues as they get older. As much as you don't want to expose the puppies and/or mom to a new environment, this is going to be necessary if this puppy continues vomiting. Dehydration can happen very quickly in little ones and death may happen within a day of the onset of vomiting.
Please let me know the status of mom and puppies in regards ***** ***** dewormer and I'll provide you with more information for your situation.