Will the Droid work with the camera removed?

I want to say no for this. It is possible that removing the camera from the printed circuit board might disturb lines and break circuit paths making the phone not work. But this is speculative at best.

If you got to remove the camera, keep a note of what pin goes where and you will want to go slowly. Make a disconnect 1 pin at a time, and test after each disconnect. Which would include powering and unpowering the phone with each pin being disconnected. Try to desolder connections rather than cut in case you need to undo any changes you make. If you discover a line/ pin that has to be connected, you could bridge that gap with very thin wire. Again this is just an educated guess on my part considering I heard very, very few demands for a camera-less Motorola droid.

Alternatively you could disable the camera with opaque material such as black plastic or metal foil, but that would depend how your work environment defines a functional camera or even if they allow a non functional camera at all. I hope this helps you out.