For the target I'm more thinking of a mechanical switch, e.g. a push button that is easy enough to press that the impact of a nerf dart can do this.
Or maybe a piezo impact sensor?
Vibration sensors don't sound too suitable, as they will vibrate from impact elsewhere on the board.
Or a light sensor, the moment the dart hits it the light is dimmed. A photodiode or phototransistor may work for this (LDRs not, too slow).