Do you Punish a five year old for...?

There HAS to be some other dynamics at play here, of which you are probably unaware. The vomiting thing is a pive-aggressive way to gain the attention (even though it's negative attention) of his parents (particularly mom). The negative attention can be just as reinforcing as the positive attention. It's important not to get stuck in the content of the behavior. Ask yourself why he might be doing this. What need is he serving? It's concerning that a 5-year old has access to all of these food items which cause him to vomit. Is there some sort of neglect happening? Abuse? My suggestion is that she spend more time with him and focus on rewarding the positive behaviors. I have to say again though, that there's probably more going on than meets the eye.