To submit a Zillow application, all required fields have to be filled out. You can add an email or employer verification website to the field. If you do not have either, you can add anything else that may help your landlord get in contact with your previous landlord or employer.
Similarly, if you do not have the phone number of a landlord or employer, you can enter a null phone number or your own phone number. You can then explain the situation directly to the landlord. You can find landlord's contact information at the top of your application.