A New York rental application is a legal document that allows a landlord to review a potential tenant's background information and references. Applicant Information.Name: SSN: Phone: Current address: City: State: Zip: Rental History – Your rental history for the last 5 years. How to fill out the New York Rental Application Form? 1. Gather all required documents and personal information. 2. Vet prospective tenants with a customized and attorneydrafted rental application for New York. Free download of an official rental application form for residential leases in New York. How to fill out the New York Rental Application Form? 1. Gather all required documents and personal information. 2. A New York rental application form is a crucial document that potential tenants must fill out in order to rent.