We study the impact of cluster environment on the morphology of a sample of 4304 extended radio galaxies from Radio Galaxy Zoo. A total of 87% of the sample lies within a projected 15 Mpc of an optically identified cluster. Brightest cluster galaxies (BCGs) are more likely than other cluster members to be radio sources, and are also moderately bent. The surface density as a function of separation from cluster center of non-BCG radio galaxies follows a power law with index -1.10 ±0.03 out to 10 r 500 (∼7 Mpc), which is steeper than the corresponding distribution for optically selected galaxies. Non-BCG radio galaxies are statistically more bent the closer they are to the cluster center. Within the inner 1.5 r 500 (∼1 Mpc) of a cluster, non-BCG radio galaxies are statistically more bent in high-mass clusters than in low-mass clusters. Together, we find that non-BCG sources are statistically more bent in environments that exert greater ram pressure. We use the orientation of bent radio galaxies as an indicator of galaxy orbits and find that they are preferentially in radial orbits. Away from clusters, there is a large population of bent radio galaxies, limiting their use as cluster locators; however, they are still located within statistically overdense regions. We investigate the asymmetry in the tail length of sources that have their tails aligned along the radius vector from the cluster center, and find that the length of the inward-pointing tail is weakly suppressed for sources close to the center of the cluster.