Delivering Essential Protection, Opportunity, and Security for Tenants Act or DEPOSIT Act This bill provides access to rental security deposit and moving cost assistance for participants in certain low-income housing programs. Specifically, the Department of Housing and Urban Development (HUD) must provide public housing agencies (PHAs) funding for rental security deposit assistance for families in Section 8 tenant-based rental assistance (TBRA) programs (i.e., the Housing Choice Voucher Program). Upon tenancy termination, any unused security deposit amounts must be returned to a PHA to remain available for use as security deposits. HUD may also provide funding to PHAs to provide assistance with reasonable moving costs for families moving into an assisted dwelling unit. Under current law, security deposit assistance (i.e., loans or grants) is available in the HOME Investment Partnerships Program for families participating in a TBRA program. This bill expands access to security deposit assistance by allowing participating jurisdictions to provide these loans or grants to very low- and low-income families regardless of participation in a TBRA program. In addition, HUD must study and report on (1) alternatives to rental housing security deposits, including requiring a tenant to pay a monthly insurance premium; and (2) the impact that such alternatives have on the rental housing market and tenants, including how such alternatives affect tenants based on tenant demographics (e.g., race, age, sex, and disability).