Keep the Watchdogs Running Act This bill authorizes a federal office of inspector general (OIG) to continue performing its duties during a lapse in appropriations (i.e., government shutdown). During a lapse in appropriations, the OIG may obligate funds (1) at the rate of operations and under the terms and conditions provided in the most recently enacted appropriations act, and (2) to perform its duties with respect to any program that continues during the lapse in appropriations.