Future space missions require that spacecraft have the capability to autonomously navigate non-cooperative environments for rendezvous and proximity operations (RPO). Current relative navigation filters can have difficulty in these situations, diverging due to complications with data association, high measurement uncertainty, and clutter, particularly when detailed a priori maps of the target object or spacecraft do not exist. The goal of this work is to demonstrate the feasibility of random finite set (RFS) filters for spacecraft relative navigation and pose estimation. The approach is to formulate satellite relative navigation and pose estimation as a simultaneous localization and mapping (SLAM) problem, in which an observer spacecraft seeks to simultaneously estimate the location of features on a target object or spacecraft as well as its relative position, velocity and attitude. This work utilizes a filter developed using the framework of RFS which are well suited to multi-target SLAM operations, avoiding data association entirely. Relevant RPO scenarios with simulated flash LIDAR measurements are tested with a Probability Hypothesis Density (PHD) RFS filter embedded in a particle filter to obtain a feature map of a target and a relative pose estimate between the target and observer. Preliminary results show that an RFS-based filter can successfully perform SLAM in a spacecraft relative navigation scenario with no a priori map of the target. These results demonstrate the feasibility of RFS filtering for spacecraft relative navigation and motivate future studies which may expand to tracking space objects for space situational awareness, as well as relative navigation around small bodies.