In this paper we present observer and output-based controller design methods for linear complementarity systems (LCS) employing a passivity approach. Due to various inherent properties of LCS, such as the presence of state jumps, mode dynamics described by differential and algebraic equations (DAEs), and regions for certain modes being lower dimensional, various observer and control design schemes that have been proposed for other classes of (hybrid) dynamical systems do not apply to LCS. In particular, we present an observer design method for LCS which is effective even in the presence of state jumps.We show the well-posedness of the observer, in the sense of existence and uniqueness of solution trajectories for the estimated state, and prove the global exponential stability of the observation error. These two properties guarantee that the estimated state exponentially recovers the state of the system. For the problem of stabilization based on output measurements only, we adopt an observer-based control approach in which we apply a state feedback law to the estimated state obtained from the observer. We prove that the resulting closed-loop system is well-posed and globally exponentially stable. In order to show the well-posedness of the closed loop, novel well-posedness results for LCS based on low-index properties are presented.