Hello I am Dr RB and I would be happy to help answer your questions/concerns about Charlie. I'm so sorry that Charlie isn't feeling well. What is Charlie's vaccinations history and the history of the puppy that you recently adopted? What is the age of the newly adopted puppy?
Any symptoms related to the gastrointestinal tract does raise concerns about transmissible (usually viral or parasitic) infections. The one of most concern would be the potential of Parvoviral enteritis. Early symptoms include mild lethargy, decreased appetite and soft stools which progresses to vomiting/diarrhea (usually with blood in stool), severe lethargy/weakness and dehydration. Without proper diagnosis and treatment (involving hospitalization) Parvo virus can be fatal. Symptoms usually occur with exposure to virus within 3-10 days of the first encounter..Parvo virus is exceptional contagious and can be transmitted via any interactions as well as via surface contact (the virus survives well in the environment on floors, in ground etc to very severe temperatures).Puppies are often carriers (even if not exhibiting symptoms) due to the weaker immune symptoms or if vaccinations are not complete to provide immunity (which isn't possible unless last vaccine was given at 16 weeks or older).
Another less concerning reason for Charlie's symptoms may be the transmission of intestinal parasites such as roundworms, hookworms, whipworms, Giardia, tapeworms to name a few. Is Charlie on a monthly preventive that helps protect him against these? The new puppy also needs to evaluated and treated for this.
Charlie's vomiting may also be attributed to conditions unrelated to the puppy such as ingestion of foreign objects or any dietary indiscretion causing an intestinal blockage and/or inflammation. This is especially common in puppies since they are so curious and like to put everything in their mouths!
Due to Charlie's young age and recent exposure to puppy with uncertain history, I would recommend that you make an appointment for Charlie with your regular veterinarian. Any signs of vomiting (especially in a young dog with the history you described) can be a sign of a gastrointestinal emergency. Given this history, your veterinarian will likely recommend a combination of diagnostics which may or may not include a fecal flotation, fecal smear, Parvo snap test, blood work and abdominal x-rays. He/she can also evaulate Charlie's hydration status and determine severity of symptoms.
I hope this advice was helpful to you. Please let me know if you need clarification on anything or have other questions about this. Also, please do not forget to rate my answer which I hope you found to be "excellent." You may do this by clicking "rate our conversation" below (Your rating is the only means of compensation I receive for answering your question). Again, let me know if I can help you further:)