Due to their ability to move without sliding relative to their environment, soft growing robots are attractive for deploying distributed sensor networks in confined spaces. Sensing of the state of such robots would also add to their capabilities as human-safe, adaptable manipulators. However, incorporation of distributed sensors onto soft growing robots is challenging because it requires an interface between stiff and soft materials, and the sensor network needs to undergo significant strain. In this work, we present a method for adding sensors to soft growing robots that uses flexible printed circuit boards with self-contained units of microcontrollers and sensors encased in a laminate armor that protects them from unsafe curvatures. We demonstrate the ability of this system to relay directional temperature and humidity information in hard-to-access spaces. We also demonstrate and characterize a method for sensing the growing robot shape using inertial measurement units deployed along its length, and develop a mathematical model to predict its accuracy. This work advances the capabilities of soft growing robots, as well as the field of soft robot sensing.