I am sorry to hear that he was vomiting and is now lethargic, and refusing to eat.
He is likely dehydrated because he has been vomiting, and not eating normally.
In most cases vomiting is triggered by eating something they should not, too much table food, too many treats or something they find outdoors.
More serious causes of vomiting include toxin ingestions, viral or bacterial infections, a dietary allergy or sensitivity, inflammatory bowel disease, internal organ failure (congenital kidney or liver disease), or a full or partial gastrointestinal obstruction.
In a young dog, especially if he never finished his vaccine series, a viral infection such as Parvo virus, toxin ingestion, or a foreign body leading to a partial or full gastrointestinal obstruction would be the most likely cause of his symptoms.
Worms may cause diarrhea, but they should not cause vomiting or a loss in appetite, and some wormers cause nausea. I don't recommend them in situations like this without a clear diagnosis. Of course it is too late now, as it has already been given, but don't give any more.
Likewise Pepto Bismol is better for diarrhea than nausea. It has salicylates which can irritate the gut, so I don't recommend any more.
Penicillin is a narrow spectrum antibiotic. It is fine for some uncomplicated urinary and respiratory infections, and we may use it in combination with other antibiotics in dogs with liver disease. But we do not know that he has a bacterial infection, and I never recommend indiscriminate use of antibiotics, especially a single dose. That leads to resistant bacteria. Please do not give him any more.
Because this has persisted for several days and he is refusing to eat ideally he would see a veterinarian now.
If that isn't possible for whatever reason there are some things we can try at home, but if he's not improving quickly he should see a veterinarian for an examination, some diagnostic testing, intravenous fluids and injectable medication to settle his stomach.
To try and settle his stomach you can give either:
1) Pepcid ac (famotidine) at a dose of one 10mg tablet per 20 to 40 pounds of body weight every 12 hours
2) Prilosec (omeprazole) at a dose of one 20mg tablet per 40 to 80 pounds of body weight every 24 hours
These are acid reducers and may help settle his stomach and get him feeling better and hopefully get his appetite back. They can be given for several days if necessary.
In a couple hours offer him small amounts of clear broth. If he drinks too much too quickly that can lead to vomiting. You can also offer ice cubes to lick. To get some electrolytes in you can offer him a 50:50 mix of pedialyte and water.
If there is no vomiting for 12 hours offer a bland diet of 1/3 boiled, minced, white skinless chicken or boiled, lean hamburger and 2/3 boiled, white rice mixed with some low salt chicken or beef broth to make it easy to lap up and swallow. If he refuses that you can offer a little meat baby food. If he refuses both then he needs hands on veterinary care.
But if things go well and he does eat and doesn't vomit feed him the bland diet for 2 to 3 days then slowly start to mix back in his regular food, a little more at each meal. It should take about 5 to 7 days to slowly convert him back to his regular diet.
If he continues to vomit even with the acid reducers, runs a fever (more than 103F rectally), or has a lower then normal temperature (less then 99F), has a tense painful belly or if he refuses to eat he should see a veterinarian for an examination, diagnostics, injectable anti-nausea drugs intravenous fluids and supportive care.
Please let me know if you have any further questions.