I am sorry to hear that your kitty isn't feeling well, vomiting everything she eats, and very lethargic.
It's tough to know whether she is lethargic because her electrolytes are off and she is dehydrated and weak from not eating or if she is lethargic due to whatever disease process is causing her to vomit.
How long has this been going on?
She is likely dehydrated because she is vomiting, but if even water is making her vomit she likely doesn't feel like drinking.
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 in include viral or bacterial infections, a dietary allergy or sensitivity, internal organ failure (kidney or liver disease), a full or partial gastrointestinal obstruction or an abdominal mass.
In a younger kitty, especially one that hasn't finished her vaccine series, a viral infection or a foreign body leading to a partial or full gastrointestinal obstruction would be the most likely cause and both can be deadly. Not all cats with viral infections run a fever initially, sometimes they are too weak to mount a fever response.
Older kitties are more likely to be suffering from organ failure or an abdominal tumor.
If this has been going on for more than 3 days, and she cannot keep down even water ideally she would see a veterinarian now. Cats that cannot eat for more than 72 hours run the risk of developing hepatic lipidosis also known as fatty liver disease. This is caused by overwhelming the liver with fats being broken down for energy to live when they don't eat, making the liver unable to function normally.
If having her examined 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 she 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 10mg tablet per 5 to 10 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 or clear broth 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, although most cats hate the taste and won't drink it. If she refuses to drink but stops vomiting you can use a medicine dropper to force water or broth into her.
If there is no vomiting for 6 hours offer a bland diet of 2/3 boiled, minced, white skinless chicken and 1/3 boiled, white rice mixed with some low salt chicken 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 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.