Video without face

The short answer is yes; Udemy doesn't require you to include any sort of image of yourself in your course. You could have a course that is just slides and screen captures with you doing a voice-over, for example.

It might put you at a slight disadvantage as it's a little more difficult to establish a personal connection with students that way, but if you're teaching a technical topic I don't think most students care if they see the instructor or not. They just want the material.

Speaking clearly will be all the more important though, if they don't have any visual cues from you.