To the OP: you can give your puppy CET chews which are supposed to be good for their teeth and they dissolve so there's no chance of a blockage like with rawhide.
I would just like to add that while working at the vet, we always asked people to bring a fecal sample with cases of vomiting OR diarrhea. With vomiting it could be a parasite so it's always better to bring a sample. In your case I'm glad it was just the raw hide and an easy fix!