A head-mounted device may include displays for presenting images to a user. The head-mounted device may have a hand tracker for gathering hand tracking input from a user. The hand tracker may include a steerable illumination system such as a steerable infrared illuminator. Infrared light may be steered by selectively activating light-emitting diodes in an array and/or using positioners to move an infrared light source and/or lens in the illuminator. An infrared image sensor may gather images of the hands. Based on the monitored movement of a hand, the steerable illumination system may steer the infrared light to ensure that the hand remains illuminated even during hand movements.