Is it 'okay' to give him milk? Yes. Are other posters correct when they say that dairy (not just milk) products can be mucus producing? Yes. If he is having congestion or phlegm, then you may wish to omit the milk for a few days while the symptoms subside.
I wouldn't worry about the vomiting unless it is something that continues for a period of time. If this were my kid, I'd avoid both milk AND dairy products for a few days and give alternate forms of nutrition that include vitamin D and calcium. Try again in a few days or when any symptoms subside. If after this time period he is asymptomatic then there really is no reason to swear dairy products off for life (I'm not one of those OMG dairy is EVIL! people, but choosing alternate sources during a period of time is fine).
And let's not forget the following:
cheese = dairy
yogurt = dairy
(unless you pick something that is not dairy-based)