Ending Homelessness Act of 2019 This bill provides additional funding for, and otherwise addresses, assistance to homeless individuals and families. The funding provided by the bill is designated as emergency spending, which is exempt from discretionary spending limits. Specifically, the bill provides additional FY2020-FY2024 funding for the Department of Housing and Urban Development (HUD) to (1) award emergency relief grants to address unmet needs of homeless populations in jurisdictions with the highest need, and (2) provide additional incremental rental voucher assistance for individuals and families who are homeless. The bill also provides FY2022-FY2024 funding for HUD to award grants to provide outreach and coordinate services for individuals and families who are homeless or formerly homeless. Further, the bill provides additional annual funding beginning in FY2020 for the Housing Trust Fund (dedicated to affordable housing for extremely low-income and very low-income families) and incremental project-based voucher and rental assistance. During the first five fiscal years in which such additional funding is available, HUD must ensure that priority for occupancy in assisted units is given to individuals and families who are homeless. Further, units receiving such assistance may not require families to contribute more than 30% of their adjusted income towards rent. Additionally, the bill provides FY2020 funding for HUD to provide technical assistance to states, local governments, and nonprofit organizations to integrate and coordinate homeless assistance with federal health care programs. Finally, the bill permanently reauthorizes certain homeless assistance grants and makes permanent the U.S. Interagency Council on Homelessness.