Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry that you have been waiting for a response, but your requested expert isn't online which delayed your question coming up on the list for all to answer. I would like to help if you are still interested in an opinion.
I am sorry to hear that Faust has been vomiting bile intermittently and for the past two days is gagging and has vomited a few times.
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), Addison's disease (hypoadrenocorticism), a full or partial gastrointestinal obstruction or even infiltrative cancers such as lymphoma (unlikely at his age).
Rarely a parasite called a stomach worm (Physaloptera) can lead to intermittent vomiting. These are difficult to diagnose because they rarely shed eggs, so we can try simply worming him with a gentle wormer called Pyrantel and see if that helps. This is easy to do and inexpensive so well worth trying.
Today it may also be helpful to put him on an acid reducing medication as too much stomach acid, especially on an empty stomach, is very irritating and predisposes to vomiting.
I recommend giving him a dose before bed so that it is in his system in the morning. You can give 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 40 to 80 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 long term 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. To get some electrolytes in you can also offer her 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 more palatable, and get additional fluids into him. If he refuses that you can offer a little meat baby food. If he refuses both then don't push it.
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.
Long term you might also consider feeding a low residue, low irritant easy to digest food as a permanent diet given this is a repeated issue. Both Science Diet and Royal Canin make sensitive stomach formulas. Some dogs seem to vomit more on an empty stomach so feeding him small meals several times a day, and maybe a snack before bedtime, may be very helpful.
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.