Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am very sorry to hear about your fellow's vomiting partially digested food several hours after eating as as some white, thin rubber pieces, but I am pleased to hear that he seems to feel pretty well.
Vomiting/retching in a younger dog can be from several causes. Common ones are a quick change in diet, too many treats, especially fatty or spicy treats, eatings things they should not such as things found in the yard, toy pieces or garbage, viral or bacterial or parasitic infections.
More serious causes include chronic pancreatitis, esophageal reflux, a dietary allergy or sensitivity, inflammatory bowel disease, internal organ failure (kidney or liver disease), a full or partial gastrointestinal obstruction or even infiltrative cancers such as lymphoma (not likely at his young age).
Given that he seems to feel pretty well a bacterial or viral infection seems very unlikely as do internal organ failure or inflammatory bowel disease. Since he is vomiting food several hours after eating that points toward slower than usual stomach emptying and the strange rubbery pieces may indicate that he found some sort of foreign body to eat which has led to a partial obstruction. Please look at his toys or around the yard for something he may have eaten.
At this point we can try some things at home but he may need to see a veterinarian if he isn't coming along well or something changes with him.
To try and settle his stomach now you can try 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 10mg tablet per 20 to 40 pounds of body weight
every 24 hours.
These will reduce stomach acid and should help if this is related to simple nausea and gastrointestinal irritation from eating something he should not have. They can be given for several days if necessary.
Starting a couple hours after the acid reducers small amounts of water or ice cubes given frequently are fine as he needs fluids after all that he has lost with vomiting.
You can give him pedialyte to replace electrolytes too.
If things are going well after he is given liquids then in a few hours start a bland diet of 1/3 boiled, lean hamburger (or boiled, white, skinless chicken), all fats and juices drained off the meat, mixed with 2/3 boiled, plain white rice. Add 1-2 tablespoons of canned pumpkin (not pie filling, just pumpkin) Feed several small meals a day. Once he feels better (no vomiting for 48 to 72 hours) start mixing in his regular dog food very slowly. Less bland more regular with each day. It should take a week or so to convert him back.
If he is continuing to vomit and starts feeling poorly though, it would be best that he see a veterinarian promptly as anything you give him orally will just come back up worsening his dehydration.
If he has a tense painful abdomen, becomes very lethargic, begins vomiting again or runs a fever greater than 103.5F then he really must be seen by a veterinarian as soon as possible.
Best of luck with your little one, please let me know should you have any further questions.