The COVID-19 pandemic has dramatically increased the use of face masks across the world. Aside from physical distancing, they are among the most effective protection for healthcare workers and the general population. Face masks are passive devices, however, and cannot alert the user in case of improper fit or mask degradation. Additionally, face masks are optimally positioned to give unique insight into some personal health metrics. Recognizing this limitation and opportunity, we present FaceBit: an open-source research platform for smart face mask applications. FaceBit's design was informed by needfinding studies with a cohort of health professionals. Small and easily secured into any face mask, FaceBit is accompanied by a mobile application that provides a user interface and facilitates research. It monitors heart rate without skin contact via ballistocardiography, respiration rate via temperature changes, and mask-fit and wear time from pressure signals, all on-device with an energy-efficient runtime system. FaceBit can harvest energy from breathing, motion, or sunlight to supplement its tiny primary cell battery that alone delivers a battery lifetime of 11 days or more. FaceBit empowers the mobile computing community to jumpstart research in smart face mask sensing and inference, and provides a sustainable, convenient form factor for health management, applicable to COVID-19 frontline workers and beyond.