Putting Our Veterans Back to Work Act of 2019 This bill addresses programs and policies related to the employment of veterans. Specifically, the bill reauthorizes through December 31, 2021, the veterans retraining assistance program, which previously provided assistance for programs of education that led to in-demand jobs. The bill authorizes up to 50,000 veteran participants in the program during FY2020 through December 31, 2023. Under the bill, the program entitles veterans for an aggregate of 12 months retraining assistance. The bill also reauthorizes through (1) March 31, 2022, additional Department of Veterans Affairs (VA) programs for certain disabled veterans who have completed a VA rehabilitation program and have exhausted their rights to state unemployment benefits; and (2) FY2023, the collaborative veterans' training, mentoring, and placement program. The VA must develop a single, unified federal web-based employment portal for veterans to access information on federal programs and activities concerning veteran employment, unemployment, and training. The bill requires the (1) Department of Homeland Security to award grants to hire veterans as firefighters, and (2) Department of Justice to award grants to hire veterans as law enforcement officers. Finally, the bill requires the Office of Federal Procurement Policy within the Office of Management and Budget to establish a preference for the procurement of certain contracts (including defense contracts) valued at $25 million or higher. Specifically, the office must establish a preference for contracting with offerors that employ a specified percent of veterans as full-time employees.