Awwww… poor kitty. As the kit been tested for parasites? Does the kit have diarrhea too, or just vomiting? If diarrhea is involved, it could be worms making the poor kitty sick.
There can be some really nasty metabolic imbalances that come from vomiting for too long. I hope the shelter does not take the kit back, because he sounds like he could really use your love and tlc. What has always worked for me with real bad vomiting is this: Start out with a 24 hour clear liquid diet to allow the stomach time to heal (ex, only pedialyte and water and broth), THEN advance to a bland diet for the next 2 to 3 days (ex, rice and chicken), THEN advance to soft foods (ex, moistened dry food or high protein canned cat food), THEN regular diet for kitty.
I know rice has little nutritional value for cats, but it is meant to allow time for the stomach to heal so the vomiting will stop, so that is okay. It is okay to continue pedialyte for the benefit of electrolytes even when the kit advances to bland foods- I always do that. The vomiting always stopped for me by the third day when using this diet. Just make sure you do not use canned broth, because it has garlic and onion powder in it which is poisonous to cats. Always check for that in broths. If you cannot find broth that does not have this, then you can make your own by boiling chicken in water & giving kitty the water.
I hope kitty gets better soon. Sending lots of love his way, please let us know how he is doing.