A New York rental application is a legal document that allows a landlord to review a potential tenant's background information and references. How to fill out the New York Rental Application Form?1. Gather all required documents and personal information. 2. Where do you fill out a renter's application? There are a few ways to fill out a rental application, depending on the landlord or property owner's preference. Learn more about New York rental applications and download our free PDF form. Screen prospective tenants with Avail landlord software. Criminal background checks are often used as screening criteria for rental housing to determine qualified applicants.