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Predicting the electrical properties of organic molecular crystals (OMCs) is challenging due to their complex crystal structures and electron-phonon (e-ph) interactions. Charge transport in OMCs is conventionally categorized into two limiting regimes − band transport, characterized by weak e-ph interactions, and charge hopping due to localized polarons formed by strong e-ph interactions. However, between these two limiting cases there is a less well understood intermediate regime where polarons are present but transport does not occur via hopping. Here we show a many-body first-principles approach that can accurately predict the carrier mobility in OMCs in the intermediate regime and shed light on its microscopic origin. Our approach combines a finite-temperature cumulant method to describe strong e-ph interactions with Green-Kubo transport calculations. We apply this parameter-free framework to naphthalene crystal, demonstrating electron mobility predictions within a factor of 1.5−2 of experiment between 100−300 K. Our analysis reveals that electrons couple strongly with both inter- and intramolecular phonons in the intermediate regime, as evidenced by the formation of a broad polaron satellite peak in the electron spectral function and the failure of the Boltzmann equation. Our study advances quantitative modeling of charge transport in complex organic crystals.