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We study the interplay between strong electron-electron and electron-phonon interactions within a two-orbital molecule coupled to metallic leads, taking into account Holstein-like coupling of a local phonon mode to the molecular charge as well as phonon-mediated interorbital tunneling. By combining canonical transformations with numerical renormalization-group calculations to address the interactions nonperturbatively and on equal footing, we obtain a comprehensive description of the system's many-body physics in the antiadiabatic regime where the phonons adjust rapidly to changes in the orbital occupancies and are thereby able to strongly affect the Kondo physics. The electron-phonon interactions strongly modify the bare orbital energies and the Coulomb repulsion between electrons in the molecule and tend to inhibit tunneling of electrons between the molecule and the leads. The consequences of these …