Veterans Back to Work Act of 2019 This bill amends the Internal Revenue Code to (1) make permanent the work opportunity tax credit for hiring qualified veterans (veterans receiving compensation for a service-connected disability and other federal assistance); and (2) allow employers who hire qualified veterans to elect to claim the tax credit as an exemption from employment and railroad retirement taxes for such veterans' first-year wages, subject to specified limitations. The bill appropriates amounts to the Social Security trust funds and the Social Security Equivalent Benefit Account to cover any revenue loss to such funds resulting from this bill.