Emergency Rental Assistance and Rental Market Stabilization Act of 2020 This bill authorizes the Department of Housing and Urban Development (HUD) to award additional grants under the Emergency Solutions Grant program to provide rental assistance for short-term and medium-term housing to homeless individuals or families, or those who are at risk of homelessness, and expands eligibility for such additional assistance. For assistance provided under the bill, the term at risk of homelessness shall include individuals and families that have an income below 80% of the median income for the area (rather than 30% of the area median income, as under current law). Rental assistance provided under the bill shall generally be limited to 120% of the HUD-established fair market rent for the area, whereas currently rental assistance under the program is generally limited to 100% of an area's fair market rent. Such assistance may not be conditioned on any prerequisite activities, such as receiving treatment. HUD may, with some limitations, waive requirements or establish alternative requirements related to such assistance if doing so is necessary to respond to the COVID-19 (i.e., coronavirus disease 2019) public health emergency.