Why does my dog vomit after eating treats?

Stop feeding treats obviously and take the dog to a vet to rule out any medical issues first. Over treating may be unbalancing the current diet, or contain ingredients that she is having an allergic reaction to.

Treats are not even a necessity, a healthy diet fed properly, and nothing else other than water that should be available free choice 24/7 that is specifically formulated for a canine that complete/balanced, fed on a set schedule, twice daily am/pm, set amount according to body condition/activity, adjusting only as needed depending on body condition, no sudden or frequent diet change, unless a vet recommends otherwise due to a specific health issue, all diet change should be done gradually as well, over a period of ten days to two weeks, by mixing old food with new food, not doing so may cause intestinal upsets, split equally into those two meals, no free feeding, human foods, scraps,etc.

Basically garbage in, garbage out. We use a well-known brand biscuit treat, and other treats from Fish4Dogs - dried fish. You could make up some baked lamb liver.

https://www.fish4dogs.com/categories/dog-shop/treats.aspx We buy the Sea Tiddlers.

Dog treats are full of disgusting, undigestible crap

"Packaged treats"doesn't tell us what kind of food it is. There are dozens of kinds of "packaged treats'. And you didn't say how long after eating them he vomits. Therefore, no one can say why the dog vomits after eating them.

No idea

Yes, probably a sensitive stomach.
There are lots of different kinds of treats made with a wide range of ingredients including binders and all sorts of stuff I can’t even pronounce.

Just like I avoid junk food, I read up on what I treat my dogs with.

Cos treats are in general rubbish, and your dog may have a weak stomach also.