Short answer is yes.
The magazines only function it the operation of a pistol is to house the ammo for future use. And has some role in the last round hold open of the slide.
Having said this there are exceptions. Some pistols have a magazine safety in them. That is, when the magazine is removed it disables the trigger system and prevents to gun from firing.
Barring the mag safety, any gun should fire without any problems but the slide may not catch and stay back.
Well. After watching his video, I have never had problems that I can remember when shooting without the mag. So please allow me to change my comment to "theoretically yes"