Hi Rafaela
1. You make me feel like I might vomit. - This is usually said to tell someone you highly disapprove of something they have done or can be said if someone makes you feel repulsed and you don't want to look at them.
2. You make me vomit - very similar to the previous but emphasises more how disgusted you are.
I have not heard 3 or 4 before, they do not sound natural to me, if you want to provide more context, I will reply again.
5. I'm not well because of you or I am ill because of you. For example if someone with a disease passed a disease to me, I would be sick because of that person.
In British English you hear 'sick' or 'to be sick ' to mean vomit. But I believe in American English it's more common to say you're 'sick' if you have any kind of illness, for example someone with a cough may say they are sick.
This is me replying to your question as a native English speaker, I'm no expert in the language. Experts may say different :)