I recommend that you Shadow to a vet for a fecal exam, some blood work and radiographs. Intestinal parasites, hairballs, obstruction, foreign bodies, organ dysfunction (liver, kidneys
, thyroid, adrenals), toxins, bacterial or viral infections and food
allergies can cause vomitting. At 7 weeks, the kittens should no longer be getting milk. They should be eating kitten food (canned and/or dry food). It could be the milk and rice cereal that is upsetting his stomach.
Until you can get him to the vet, don't give him any food for 12 hours. He can have water but don't let him overdo it. If he can't keep the water down then you'll need to get him to the vet where they can put him on IV fluids. After 12 hours offer him small amounts of a bland diet like boiled unsalted unseasoned skinless chicken breast (or tuna in water) and some white rice (1 part meat to 1 part rice). Offer him a small amount (1 tablespoon) and if he doesn't vomit after 3 hours, give him some more. You can feed him 3 small portions that day. The next day, feed him a normal amount of food split into 4 meals. Continue that for 3 days before returning to his regular feeding schedule and food.
If the vomitting does not improve in a day or so, if he cannot keep anything in his stomach, if he becomes lethargic or anorexic, develops diarrhea
or if he seems to be in a lot of pain then get him to the vet as soon as you can.
I hope this helps.