VOW to Hire Heroes Extension Act of 2019 This bill amends the Internal Revenue Code to (1) make permanent the work opportunity tax credit, including with respect to qualified veterans; (2) revise tax credit eligibility requirements for documenting the status of veterans and their receipt of unemployment compensation; and (3) extend the payroll tax offset for such credit to certain for-profit employers. The Internal Revenue Service must make annual reports to Congress on the effectiveness and cost-effectiveness of this bill in increasing the employment of veterans. The bill directs the Department of the Treasury to pay (1) each U.S. possession (i.e., American Samoa, Guam, the Commonwealth of the Northern Mariana Islands, the Commonwealth of Puerto Rico, and the U.S. Virgin Islands) with a mirror code tax system amounts equal to the loss to such possession due to this bill; and (2) each U.S. possession without such a tax system an amount estimated to equal the loss to such possession that would have occurred due to this bill if such a tax system had been in effect in that possession.