In most cases vomiting is triggered by eating something they should not, too much table food, too many treats or something they find outdoors (like fertilizer), or an abrupt diet change.
More serious causes of vomiting in puppies include viral or bacterial infections, a dietary allergy or sensitivity, congenital internal organ failure (kidney or liver disease), or a full or partial gastrointestinal obstruction.
In a young dog, especially one that hasn't finished her vaccine series, a viral infection such as parvo virus or a foreign body leading to a partial or full gastrointestinal obstruction are big concerns. Not all dogs with viral infections run a fever initially, sometimes they are too weak to mount a fever response. Worms can cause loose stools, but rarely cause vomiting and lethargy.
Because she is young, this continues for more than a day, and she is lethargic ideally she would see a veterinarian now. Young puppies dehydrate easily and she needs fluids.
If that isn't possible for whatever reason there are some things we can try at home, but we cannot replace in clinic intravenous fluids and injectable medications so if she isn't responding quickly he should see a veterinarian promptly.
To try and settle her stomach you can give either:
1) Pepcid ac (famotidine) at a dose of one quarter of a 10mg tablet per 5 to 10 pounds of body weight every 12 hours
2) Prilosec (omeprazole) at a dose of one quarter of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours
These are acid reducers and may help her feel less nauseous and hopefully stop the vomiting and improve her appetite. They are quite safe and can be used for several days if necessary.
I would pick up all food for now and water for a couple hours to allow her stomach to settle after the acid reducers.
In a couple hours when you give her water make sure it is in small amounts only. If she drinks too much too quickly that can lead to vomiting. To get some electrolytes in you can also offer her a 50:50 mix of pedialyte and water.
If there is no vomiting for 6 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 and get additional fluids into her. If she refuses that, you can offer a little meat baby food. If she refuses both then don't push it, she needs hands on veterinary care as soon as possible.
But if things go well and she does eat the bland diet and doesn't vomit feed her the bland diet for 3 to 4 days then slowly start to mix back in her regular puppy food, a little more at each meal. It should take about 5 to 7 days to slowly convert her back to her regular diet.
If she continues to vomit even with the acid reducers, runs a fever (more than 103F rectally), or has a lower than normal temperature (less than 99F), has a tense painful belly, or if she refuses to eat even after the acid reducer is given she 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.