Polling Access Safety Act of 2019 This bill limits immigration enforcement in relation to polling places and establishes various requirements related to the Department of Homeland Security (DHS). Absent exigent circumstances, the following are prohibited on a voting day: (1) immigration enforcement actions within one mile of a polling place; (2) identification checks at a polling place by a federal officer or employee; and (3) DHS-conducted temporary vehicle checkpoints, public transportation searches, and public training exercises. State election administrations shall post notice of such prohibitions at each polling place on a voting day. Information acquired in violation of such prohibitions may not be used to initiate deportation proceedings. DHS shall establish a training program for certain officers and employees on compliance with federal laws limiting certain political activities of federal employees and officers (e.g., the Hatch Act). The bill establishes three positions to monitor and ensure compliance with the bill. These positions shall be within the DHS Office of Inspector General, the DHS Office for Civil Rights and Civil Liberties, and the Office of Special Counsel.