The honeycomb lattice of graphene is a unique two-dimensional (2D) system where the quantum mechanics of electrons is equivalent to that of relativistic Dirac fermions. Novel nanometer-scale behavior in this material, including electronic scattering, spin-based phenomena, and collective excitations, is predicted to be sensitive to charge carrier density. In order to probe local, carrier-density dependent properties in graphene we have performed atomically-resolved scanning tunneling spectroscopy measurements on mechanically cleaved graphene flake devices equipped with tunable back-gate electrodes. We observe an unexpected gap-like feature in the graphene tunneling spectrum which remains pinned to the Fermi level (E_F) regardless of graphene electron density. This gap is found to arise from a suppression of electronic tunneling to graphene states near E_F and a simultaneous giant enhancement of electronic tunneling at higher energies due to a phonon-mediated inelastic channel. Phonons thus act as a "floodgate" that controls the flow of tunneling electrons in graphene. This work reveals important new tunneling processes in gate-tunable graphitic layers.