Navigation planning for legged robots has distinct challenges compared to wheeled and tracked systems due to the ability to lift legs off the ground and step over obstacles. While most navigation planners assume a fixed traversability value for a single terrain patch, we overcome this limitation by proposing a reachability-based navigation planner for legged robots. We approximate the robot morphology by a set of reachability and body volumes, assuming that the reachability volumes need to always be in contact with the environment, while the body should be contact-free. We train a convolutional neural network to predict foothold scores which are used to restrict geometries which are considered suitable to step on. Using this representation, we propose a navigation planner based on probabilistic roadmaps. Through validation of only low-cost graph edges during graph expansion and an adaptive sampling scheme based on roadmap node density, we achieve real-time performance with fast update rates even in cluttered and narrow environments. We thoroughly validate the proposed navigation planner in simulation and demonstrate its performance in real-world experiments on the quadruped ANYmal.