Veteran Housing Opportunities and Unemployment Support Extension Act of 2019 This bill expands eligibility for the Department of Housing and Urban Development (HUD) Veterans Affairs-Supported Housing (VASH) rental voucher assistance to homeless veterans honorably discharged or released from active military, naval, or air service regardless of length of service. According to federal regulations, HUD was granted authority in the Consolidated Appropriations Act, 2008 to waive or provide an alternative for the current requirement that HUD-VASH assistance be targeted to homeless veterans who have chronic mental illnesses or chronic substance-use disorders.