Yes, but it will feel weird. VR is all about mimicking real life, meaning that your "head" needs to control only your head.
Nobody has a gun glued to their heads. Head movement and arm movement need to be decoupled.
How about something like this?
This is the control scheme they used in the The Deep demo.
Apparently it felt obvious and natural enough that barely anyone even mentioned it, much less really made the connection. Another connection no one seemed to make was, if they weren't trapped in that cage, they could've used the dual analog sticks to swim around.
So to address your concern, DieH@ard,
in addition to performing its normal duties as your controller, the DS4 has no trouble simultaneously standing in for your raygun/flashlight/axe/chainsaw/hand/tentacle/whatever.
I imagine that would ruin immersion as the screen in front of you would be shaking. In some ways, it's very gimmicky as well.
For example in a car crash, that would just break immersion as you won't only feel the sensation of "crashing" on the side of your face.
Add to that, it would add unnecessary weight as well as increase price. It adds nothing to the headset.
First of all, every one of those arguments could be made against putting rumble in the controller as well.
More to the point, it can be really useful to be able to give players a boot to the head. You're playing a game and some big dude knocks you on your ass. As you hit the ground, your vision blurs as you feel a thump at the base of your skull followed by a tingling that travels down your spine. As you struggle to your feet, the tingling fills your entire head before taking up residency in your temples as a high-frequency buzz that seems to follow gravity as you wobble back and forth.
Also, two words:
Spidey Sense
You could simulate all kinds of crazy senses with a reasonably sophisticated rumble system. Imagine a Dwarf that can find treasure by sensing the direction and intensity of the vibes it gives off.