Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that your fellow is vomiting hourly. Although the fact that he wants to eat and drink is a good sign, if even water is making him vomit you need to take it away from him for now. He is getting no benefit from drinking if he vomits everything he's taking in, and in fact will only dehydrate himself further.
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 at this age include toxin ingestions, viral or bacterial infections, esophageal reflux, a dietary allergy or sensitivity, inflammatory bowel disease, internal organ failure (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 gastrointestinal irritation or a partial or full gastrointestinal obstruction would be the most likely cause.
Because he has been vomiting so frequently for 24 hours and he is vomiting even water 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 half of a 10mg tablet per 10 to 20 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 settle his stomach and get him feeling better and hopefully get him to stop vomiting. They can be given for several days if necessary.
I would pick up all food for now and water for a couple hours to allow his stomach to settle after the acid reducers.
In a couple hours when you give him water make sure it is in small amounts only. 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 don't push it, he needs 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 cannot be treated at home and 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.