This sounds very concerning. In a puppy of this age my top two concerns are parvovirus or intestinal parasites. Parvovirus is a disease that we vaccinate against in the "distemper" vaccine combo in a series of 3 to 4 shots given at 3 week intervals. If your puppy is in the middle of his vaccine series then this is less likely. Puppies get it by being in an area where a dog that had the disease was. And the virus survives in the environment for up to 2 years! The vomit/diarrhea usually progresses to become bloody. It is life threatening and requires hospitalization to treat, or in almost 98% of cases the puppy will die.
Gastrointestinal parasites (roundworms, hookworms) are not quite as life-threatening but if the vomit/diarrhea is allowed to continue without addressing the problem then it can become life threatening due to secondary anemia and dehydration. Over-the-counter dewormers can help but it really does depend on the type. I would recommend panacur. Deworming, however, doesn't replace a veterinary visit. If your puppy has parvovirus, the longer you wait, the sicker she will get and the less chance we have of saving her. There is a very quick in-house test that all vet clinics have to test for parvovirus.
Now as a final two possibilities for cause of the vomit/diarrhea: abrupt diet change and gastrointestinal foreign body. Both of these may be things that you have suspicions of. And you can consult with your veterinarian as to the best way to proceed.
Best of luck. I am very concerned about your puppy.