If you feel really envious then you can get a kind of sick feeling in your stomach (or perhaps it's in your heart! Or your head!). It's not the same sort of sick feeling as when you eat too much! It's not like gastroenteritis! And I do think it's more in your head or heart than your stomach. But it's a bad feeling. You really really want something and you haven't got it, and can't have it, and you feel kind of sick about it. And I agree with Martim, you can almost hate the person who has what you want. I think there are quite a few emotions that can leave you feeling sort of sick.

"sick with envy" is a metaphor, he doesn't literally become sick, but because he was so envyous it was like he became "sick", meaning that he was very envyous, almost to a degree that he was hateful. I Hope I helped!