Hello, I'm Dr. Deb and will do my best to help you today.
First of all, congratulations on the newest member of your family and thanks for giving Koda a good home and a second chance.
If he's otherwise acting like his normal self, then I probably wouldn't be too worried about the vomiting that he's doing, especially if there's no food in it. He may have developed a mild case of gastroenteritis or esophagitis; even internal parasites such as round worms (which can migrate into the stomach) might trigger this behavior.
However, having said that, sometimes vomiting can be confused with coughing which would most likely be secondary to kennel cough given that he's a rescue.
But if you're fairly sure that he's vomiting, I do have a few suggestions which, hopefully, will minimize or resolve it:
1. Over the counter Pepcid AC (Famotidine) can be given at a dose of 1/2 of a 10 mg tablet twice a day which should help with his nausea.
2. I might feed him several small meals throughout the day as opposed to just one or two. I don't think it's absolutely necessary to feed a bland diet of boiled chicken and rice but this might be something to consider.
3. I suspect the shelter wormed him but it's not uncommon for these dogs to be rewormed again. I'd take a fecal sample with you for your vet visit on Monday.
I hope this helps. Deb