Actually it's very much possible. The light will move at c out of the laser then interact with dust and move towards the camera, again at c. Both of these things take a tiny amount of time. Imagine a virtual triangle made by you at one point with the camera and the laser emitting at the other end (one point is the camera, another point is the beam start/laser, and third is the end of the beam).
However, if you increase these lengths, then there is definitely a way to film an emerging beam. You would need an obscenely fast camera OR an obscenely large distance, or a little bit of both.