The ability of two nearly-touching plasmonic nanoparticles to squeeze light into a nanometer gap has provided a myriad of fundamental insights into light-matter interaction. In this work, we construct a nanoelectromechanical system (NEMS) that capitalizes on the unique, singular behavior that arises at sub-nanometer particle-spacings to create an electro-optical modulator. Using in situ electron energy loss spectroscopy in a transmission electron microscope, we map the spectral and spatial changes in the plasmonic modes as they hybridize and evolve from a weak to a strong coupling regime. In the strongly-coupled regime, we observe a very large mechanical tunability (~250 meV/nm) of the bonding-dipole plasmon resonance of the dimer at ~1 nm gap spacing, right before detrimental quantum effects set in. We leverage our findings to realize a prototype NEMS light-intensity modulator operating at ~10 MHz and with a power consumption of only 4 fJ/bit.