I would definitely try to buy him the same food he's been eating if you run out. Switching low quality foods (ones found in a grocery store) can really cause an upset system. Much worse than him vomitting.
If the vomit was yellow, a little foamy and slimy, then he threw up due to an empty stomach. The stomach acids didn't have any "work" to do.
In order and one at a time I would do:
Try wetting his food with water. Even if it's not a "gravy style" food, wetting it seems to be some kind of treat for them. Sometimes.
Also, some dogs don't like to eat in front of an audience. Can you put his food with him at night, leave it out where he can get to it, something along those lines?
My next step would be adding a little broth to it. Just enough to soak into the food and get his nose in the bowl. Chicken broth from a can is OK.
Even though table scraps are much better for him than a lot of dog foods on the market, his owners may not appreciate it. He may become a table beggar when they are eating, if he isn't already. But YT is right, dogs existed for years before commercial foods became available and they suddenly started needing all these vaccinations.... but that's a whole nuther topic
It's really hard with him being passed around, but I wouldn't want to send him home with any bad habits he may or may not pick up. If the owners even care.
And, dogs can go 48 hours without even feeling hungry. Most dogs eat out of habit and because they enjoy it. Although knowing that may not stop the yellow bile. It's just that he isn't going to die or anything if he isn't eating every day.... unless he's diabetic that is.