I suppose the closest approach to taking a video recording of moving light can be realized by choosing an approach similar to what xxChrisxx suggested:
Take a short laser beam and manage somehow to get some of the light reflected at an well defined angle over a length of 3 mm or so and then shine this line of 3 mm length on the entrance slit of a streak camera. Inside the camera these photons will hit a photocathode and be converted to photoelectrons. Now you can accelerate them towards some screen which will glow when hit by electrons (usually some phosphor screen) and at the same time apply some sinusoidal or ramped vertical deflection voltage, so that the position where an electron hits the phosphor screen will depend on the time when it hit the photocathode. Now just take a picture of the screen afterglow.
You will get a picture that shows the position where the light hit the photocathode (corresponding to the position where it was when it was reflected) on the horizontal axis and that shows the time when the photon arrived at the photocathode on the vertical axis. So if you do it correctly, you should get a diagonal line moving from the top left to the bottom right, correlating arrival position and time. The possible timing resolution in streak cameras is on the order of 500 fs to 1.5 ps. Typical photocathode sizes are about 3 mm (corresponding to roughly 10 ps travel time at the speed of light), so it should be possible to monitor the movement of a short light pulse. However, this is of course just monitoring, not taking a direct picture.