I am sorry to hear that Millie is vomiting yellow, bile material in the early mornings.
Yellow in the vomit means that the small intestine is refluxing bile into the stomach so that when she 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.
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 her with a gentle wormer called Pyrantel and see if that helps. This is easy to do and inexpensive so well worth trying.
If that doesn't work and since she seems to feel well otherwise some dogs seem to vomit more on an empty stomach so feeding her small meals several times a day, and maybe a snack before bedtime, may be very helpful.
It may also be helpful to put her 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 her a dose before bed so that it is in her system in the morning. You can give either:
1) Pepcid ac (famotidine) at a dose of one 10mg tablet per 15 to 40 pounds (7 to 20 kilos) of body weight every 12 hours
2) Prilosec (omeprazole) at a dose of one 20mg tablet per 30 to 80 pounds (14 to 40 kilos) of body weight every 24 hours
These are both acid reducers and either one could help her vomiting. They are quite safe and can be used long term if necessary.
You might also consider feeding a low residue, low irritant easy to digest food. Both Science Diet and Royal Canin make sensitive stomach formulas you can purchase at the pet store.
If she continues to vomit even with the acid reducers, runs a fever (more than 103F rectally), has a tense painful belly or if she refuses to eat she 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.
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