Gate-controlled tuning of the charge carrier density in graphene devices provides new opportunities to control the behavior of molecular adsorbates. We have used scanning tunneling microscopy (STM) and spectroscopy (STS) to show how the vibronic electronic levels of 1,3,5-tris(2,2-dicyanovinyl)benzene molecules adsorbed onto a graphene/BN/SiO2 device can be tuned via application of a backgate voltage. The molecules are observed to electronically decouple from the graphene layer, giving rise to well-resolved vibronic states in dI/dV spectroscopy at the single-molecule level. Density functional theory (DFT) and many-body spectral function calculations show that these states arise from molecular orbitals coupled strongly to carbon-hydrogen rocking modes. Application of a back-gate voltage allows switching between different electronic states of the molecules for fixed sample bias.