If there is vomiting, diarrhea and lethargy in a puppy then he should be tested for Parvo
Virus. This is a common virus in puppies that causes these signs. It is part of the distemper
vaccine, but sometimes in puppies that are unvaccinated or not fully vaccinated we'll get the disease.
The diagnosis is made with a rectal swab snap test.
Treatment is with aggressive IV fluids, IV antibiotics and GI protectants. It usually takes 3-5 days in the hospital to recover from Parvo. With aggressive treatment, > 90% of puppies will survive. With at home or out-patient care, the survival rate is only 30-40%.
I would get him tested asap either with your family vet or an ER vet that has 24 hour care as that would be best for him. Hopefully it's just a parasite or some other GI disease, but Parvo is most likely and needs to be ruled out.
I hope this helps, let me know if you have any other questions.