Suspended carbon nanotubes are known to support self-driven oscillations due to electromechanical feedback under certain conditions, including low temperatures and high mechanical quality factors. Prior reports identified signatures of such oscillations in Kondo or high-bias transport regimes. Here, we observe self-driven oscillations that give rise to significant conduction in normally Coulomb blockaded low-bias transport. Using a master equation model, the self-driving is shown to result from strongly energy-dependent electron tunneling, and the dependencies of transport features on bias, gate voltage, and temperature are well reproduced.