How to reject a tenant?

Email would be just fine. Even if Zillow had a built-in way to reject applicants, I would send an email. Someone might not have their notifications set up, or have used a burner email for their Zillow account (different then what they put on the application).

Whether you tell them why you rejected the application is up to you. If you have a justifiable reason then it may be helpful for the applicant to know why. They may have even forgot to give you some information (like a secondary income source).

Personally, we rented and then purchased a home that was originally found on Zillow and none of the communication with the landlord/realtor was actually done through the Zillow website other than the initial contact. It was all done through phone and email.