We investigate exciton transport through one-dimensional molecular aggregates interacting strongly with a cavity mode. Unlike several prior theoretical studies treating the monomers as simple two-level systems, exciton-vibration coupling is explicitly included in the description of open quantum dynamics of the system. In the framework of the Holstein-Tavis-Cummings model with truncated vibrational space, we investigate the steady-state exciton transfer through both a molecular dimer and longer molecular chains. For a molecular dimer, we find that vibration-assisted exciton transfer occurs at strong exciton-cavity coupling regime where the vacuum Rabi splitting matches the frequency of a single vibrational quantum, whereas for longer molecular chains, vibration-assisted transfer is found to occur at the ultrastrong exciton-cavity coupling limit. In addition, finite relaxation of vibrational modes induced by the continuous phonon bath is found to further facilitate the exciton transport in vibrational enhancement regimes.