We propose and experimentally demonstrate a reactive planning system for bipedal robots on unexplored, challenging terrain. The system includes: a multilayer local map for assessing traversability; an anytime omnidirectional control Lyapunov function for use with a rapidly exploring random tree star (RRT*) that generates a vector field for specifying motion between nodes; a subgoal finder when the final goal is outside of the current map; and a finite-state machine to handle high-level mission decisions. The system also includes a reactive thread that copes with robot deviations via a vector field, defined by a closed-loop feedback policy. The vector field provides real-time control commands to the robot's gait controller as a function of instantaneous robot pose. The system is evaluated on various challenging outdoor terrains and cluttered indoor scenes in both simulation and experiment on Cassie Blue, a bipedal robot with 20 degrees of freedom. All implementations are coded in C++ with the robot operating system and are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/UMich-BipedLab/CLF_reactive_planning_system</uri> .