I'm considering on making a mid budget movie and I want to get in touch with Chloë Grace Moretz. Do you know how?

That would be very difficult. Your best bet is to see if she has a fan page, and if she's a member of that page. Try contacting through email via that page. Also check fb, to see if she has a page there.

She likely gets thousands of emails a day, so even if you do find a way to contact her, don't expect her to respond.

Good luck.

If you are rich enough to be able to do a movie, you would know you do not contact the stars directly. (Unless you are super rich and then you would know how)

You contact the persons agent.