Microelectromechanical systems (MEMS) have been applied to many measurement problems in physics, chemistry, biology and medicine. In parallel, cavity optomechanical systems have achieved quantum-limited displacement sensitivity and ground state cooling of nanoscale objects. By integrating a novel cavity optomechanical structure into an actuated MEMS sensing platform, we demonstrate a system with high-quality-factor interferometric readout, electrical tuning of the optomechanical coupling by two orders of magnitude and a mechanical transfer function adjustable via feedback. The platform separates optical and mechanical components, allowing flexible customization for specific scientific and commercial applications. We achieve a displacement sensitivity of 4.6 fm Hz-1/2 and a force sensitivity of 53 aN Hz-1/2 with only 250 nW optical power launched into the sensor. Cold-damping feedback is used to reduce the thermal mechanical vibration of the sensor by three orders of magnitude and to broaden the sensor bandwidth by approximately the same factor, to above twice the fundamental frequency of ≈40 kHz. The readout sensitivity approaching the standard quantum limit is combined with MEMS actuation in a fully integrated, compact, low-power, stable system compatible with Si batch fabrication and electronics integration.