Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that your rescue puppy Hercules has vomited worms and I understand that you are concerned because he is drinking lots of water and vomiting though he has been given a wormer and his first set of vaccines.
If they are long, white spaghetti like worms they are probably roundworms.
Roundworms are very commonly found in puppies, they are easily passed from mom to pup and are probably the most common parasite in puppies.
When we worm puppies with small numbers of worms they will often be digested and we may not see any evidence of them having them, or we may see a few dying worms get passed. This is quite normal and tells us the wormer is working. In pups with large numbers of worms they may not only pass worms along with stools, but they may also vomit them. That does likely point toward your pup having lots of worms, but it is normal. If the wormer hadn't weakened the worms they would still be attached to the intestinal wall and they couldn't be vomited up. The wormer is working.
I recommend worming pups with a general wormer monthly. Using a heartworm prevention with a wormer (like Heartgard Plus) monthly is a great way to do this.
If your pup is bright and alert other then vomiting worms, and the vomiting stops then he should be fine. But if his nausea continues then he should see a veterinarian for an injection to settle his stomach.
Because Hercules is a puppy, and especially because he is a rescue fellow, this may be more than just an overwhelming number of worms. It is possible that your puppy is suffering from an infectious viral infection, for example Parvo virus. I know that he started his vaccines, but he may have been exposed before being vaccinated, and most dogs aren't fully protected until they are 4-5 months of age and their immune system matures.
If he isn't improving with my suggestions he needs to be rechecked.
To try and settle his stomach today you can try 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 will reduce stomach acid and may help settle his stomach. They are quite safe and can be used for a few days if necessary.
With-hold food and water for a couple hours and if he stops vomiting then offer a small amount of water and ice cubes to lick.
Then if he still doesn't vomit feed a homemade diet of 1/3 boiled hamburger or white, skinless chicken, all fats and juices drained and 2/3 white rice for a couple days. Once he feels well again then slowly transition him back to his regular puppy diet slowly by adding more regular and less bland at each meal.
But please be aware that it is possible that his vomiting isn't due to worms, and he may have a viral infection. If his nausea continues he needs a veterinary visit.
Please let me know if you have any further questions.