Yes, mostly. The Coulomb force is described by but one of the four Maxwell Equations, acting between the (mostly) stationary protons and neutrons in the nucleus and the fast-moving electrons that can be found, each in its orbital, though protons, neutrons, and electrons all also have magnetic moments. Coulomb's Law is the only one of the equations that is needed to describe interactions between stationary charges; you need the others for moving charges.
So you're almost correct in your hypothesis: covalent bonds are a form of electrodynamic (not electrostatic) interactions.
You could bend the shapes of the orbitals (and change the chemicals that result) by doing your experiments in an impossibly strong electric or magnetic field, but it would all still be electrodynamic interactions. To see the effect of gravity, you could run your experiments in a region of extremely high gravity (e.g., very close to the event horizon of a black hole). The nuclear strong force doesn't seem to care about electrons, and the nuclear weak force does, but only for electrons moving fast enough to smash their way out of any known chemical bond.