Hello, I'm Dr. Deb and will do my best to help you today.
I'm sorry that Ahzee isn't feeling well.
When a dog this age stops eating and is lethargic, then I always worry that something systemic may be going on, unfortunately. There doesn't necessarily have to be, of course; he may just be having an "off"day or so or he may have picked up a viral/bacterial infection but if he's been like this for more than just a few days, then I'd start to become a little concerned.
More than likely, the bile he's vomiting is because he's not eating. Bile can be quite irritating to the stomach lining and is produced even when a dog isn't eating.
Over the counter, Pepcid AC (Famotidine) can be given which should help with this acid production. The dose would be 1/4 mg per pound of body weight twice a day. For example, the dose for a 15-20 lb dog would be 1/2 of a 10 mg tablet at one time.
Unfortunately, there are no good over the counter appetite stimulants which could encourage him to eat but perhaps if he's feeling a little less nauseous, his appetite will return.
I don't know if you've also tried to tempt him with baby food but this might be something else to consider. Just avoid those with onion or garlic in them.
You could also try to force feed him although it might be a little messy. You could mix canned dog food (or the baby food I mentioned) with low sodium broth (or pedialyte) in a blender to the consistency of soup and use a syringe or small turkey baster to dribble this mixture into the side of his mouth. At least, he'll get some nutrition this way.
If he continues to vomit, however, or continues to refuse food and/or is lethargic, then a vet visit may be prudent.
I hope this helps and that he's feeling better soon. Deb