Periodic confinement of surface electrons in atomic structures or extended nanoporous molecular networks is the archetype of a two-dimensional quantum dot (QD) superlattice. Yet, an electrical control of such an artificial lattice by external gating has never been demonstrated. Here we show the capacitive coupling between an atomic force microscope (AFM) and quantum states in highly crystalline nanoporous molecular networks on Ag(111). We characterize their local density of states (LDOS) using scanning tunneling microscopy (STM). Low-temperature force spectroscopy reveals force/dissipative responses at threshold voltages that arise from the charging/discharging of the superlattice eigen-states under the local electric field of the tip. We infer the quantum capacitance and resonant tunneling rates, opening new avenues in the characterization of exotic phenomena in designer quantum materials via a nanomechanical oscillator.