Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Gaston has some chronic health issues (overlong soft palate and vomiting) and is now vomiting yellow fluid (bile) along with his breakfast and foam.
Bulldogs and other flat faced breeds are known for having a complex of issues that can make breathing more difficulty, called brachycephalic syndrome. These issues include an overlong soft palate, stenotic (small, closed in) nares, and parts of the larynx that evert out when breathing called everted laryngeal saccules. These pups will struggle to breathe and gag, and often spit up small amounts of mucous when overly stressed or excited. It sounds like that may have been what you saw in the past with him.
Vomiting foam simply is a mix of air and stomach/esophageal mucous made when he retches, it is not indicative of any disease process but tells us he is quite nauseous and retching hard.
Yellow in the vomit means that the small intestine is refluxing bile into the stomach so that when he vomits you see the yellow color. That isn't normal as bile doesn't belong in the stomach, and it does mean that there is some reverse motility, but it isn't specific for any particular disease process.
In many cases vomiting is triggered by eating something they should not, too much table food, too many treats or something they find outdoors, esophageal reflux, or a dietary allergy or sensitivity.
More serious causes of vomiting include viral or bacterial infections, chronic pancreatitis, inflammatory bowel disease, internal organ failure (kidney or liver disease), a full or partial gastrointestinal obstruction or even infiltrative cancers such as lymphoma.
How is he feeling otherwise today? Is he bright, alert and happy or is he lethargic?
If he seems to feel well otherwise it may be helpful to put him on an acid reducing medication as too much stomach acid predisposes to vomiting.
You can give either:
1) Pepcid ac (famotidine) at a dose of ¼ of a 10mg tablet per 5 to 10 pounds of body weight every 12 hours
2) Prilosec (omeprazole) at a dose of ¼ of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours
These are both acid reducers and should help him feel better. They are quite safe and can be used for several days if necessary.
I would not feed him any food for 12-24 hours after the acid reducers are started.
This should help stop gut spasms and restore normal gut motility. Small amounts of water or ice cubes given frequently are fine as he needs fluids after all that he has lost with vomiting and diarrhea. You can give him pedialyte to replace electrolytes too but Gatorade is much too high in sugar which can make intestinal irritation worse. Do not him drink too much at any one time as that can lead to vomiting.
After his food fast start a bland diet of 1/3 boiled hamburger (or boiled, white, skinless chicken), all fats and juices drained off the meat, mixed with 2/3 boiled, plain white rice. Feed small meals frequently until you figure out that it is sitting well with him, then gradually increase the amount and decrease meal frequency. I would start with 1/4 cup for the first meal.
Once he feels better (no vomiting for 48 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 this is a chronic issue for him you might also consider feeding a low residue, low irritant easy to digest food.
Both Science Diet and Royal Canin make sensitive stomach formulas.
If he continues to vomit even with the acid reducers, runs a fever (more than 103F rectally), has a tense painful belly or if he refuses to eat he should see a veterinarian for an examination, and some diagnostic testing. I would start with a complete blood count and biochemistry profile, a specific test for pancreatitis called canine specific pancreatic lipase, as well as fecal checks for parasites.
Please let me know if you have other details or a particular question based upon my response.