Yes, the camera plays an important role in Facial Recognition:
Position of the camera: Acceptable face images should be within pitch 45 deg, yaw 45 deg and roll 25 deg. So, the mounting of the camera for Video facial recognition is critical for some use-cases.
Focal length: Preferably using a standard lens rather than a wide angle lens.
Dynamic range: Wide Dynamic Range cameras tend to perform well in outdoor conditions.