Any class and any weapon can lay down suppressive fire. The amount of suppression achieved depends on 3 things:
The proximity of the bullets to the person
The size of the bullets fired (bigger bullets, and especially frag rounds and explosives, will achieve more of a suppressive effect, so an smg will have less of a suppressive effect than a sniper rifle)
The amount of bullets fired (the suppression effect continues to stack until it reaches the cap, so an smg can be just as suppressive as a sniper rifle, but will require more bullets).
All of this coupled together makes the Support class generally the best class to purposely lay down suppressive fire, as using the bi-pod will make you accurate enough to get close proximity, your bullets are large enough to achieve quite a good suppression effect per bullet and you can fire much more than any other class to reach the suppression cap much more quickly.
With that said, generally, you won't purposely try to lay down suppressive fire much (unless you're in a really good squad who actually communicates and works as a team), you'll generally be taking a few shots at an enemy and a few will miss, causing suppression, and as long as they're at least a little bit suppressed, you'll get the suppression bonus if someone else kills them (they don't need to be very suppressed at all, just enough to cause the screen blur).