VOW to Hire Heroes Extension Act of 2021 This bill (1) makes permanent the work opportunity tax credit with respect to qualified veterans, (2) revises tax credit eligibility requirements for documenting the status of veterans and their receipt of unemployment compensation, and (3) extends 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 to 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) amounts equal to the losses to such possessions due to this bill.