If you find a rental you're interested in, contact the listing owner through the contact manager box on the right hand side of the listing. You can fill out the form with your name, phone, email and a brief message, or call the property manager directly by dialing the phone number in the contact manager box. If you don't see the contact box at the top of the page, scroll to the bottom of the listing and you'll find the contact information there.
Zillow does not have direct control over the communication between a listing provider and potential tenant. If you've inquired about a listing but haven't gotten a response, you can refer to this article for tips and next steps.