My son was diagnosed with lactose intolerance in first grade. Cut out all dairy products, not just milk. Ice cream, cheese, milk chocolate, etc., can all cause stomach upsets and pain. You can get lactase in pill form at most general health/ nutrition centers . If they take these with the dairy product, they can usually handle it ok. It can get them through birthday and pizza parties. Educate your child on what they cannot eat. Substitute yogurt products for dairy products in order to insure proper calcium intake. Your child is young enough to learn to like them, if they don't already, especially frozen yogurt.
Lactose intolerance is usually the result of one's body not producing enough lactase to break down the chemical sugar in dairy products. Incidentally, onions and mushrooms contain sugars that are very similar to dairy product sugars, so they, too, might cause stomach upsets in your child.