How do I get the email of the one submitting contact form in the "reply to"?
If the person submitting the form is an anonymous visitor the form will be empty and the visitor will have to fill in every field. 

If the visitor is logged in (authenticated user) the name and email address are already pre-filled from their user profile. 

Are you sure you're not testing this while being logged in as an admin account that has the same email address as the site? Then it's expected to see that email there. 

Try to login as a different user, and also test as anonymous to see that the email is now from that different user/empty respectively for those two cases. 

If you're constantly seeing the site email as From in all three cases then it's a bug and you should try to see where it's coming from. I haven't seen any problems with the Contact module, it's showing the correct "From" value every time.