No, not really.
To penetrate something is really about energy density, you have to have enough joules for the area of impact for the desired effect. This means that a penetrating weapon needs to have a good energy density and be efficient in transferring that energy to the target.
Kinetic energy density depends on the mass density and square of velocity. Unfortunately the density of the projectile of a vortex cannon is for any significant distance the same as the ambient air density, which is to say very low both in absolute terms and in comparison to human body. Similarly the speed is capped well below the speed of sound or you would in effect be building a sonic shock-wave weapon not a vortex gun. (This might actually be a useful weapon though; shock waves have lots of interesting physical effects.)
This means that beyond certain value you can only increase the kinetic energy of the impact by making the vortex bigger which does absolutely nothing to increase energy density. And since the density of air is quite low compared to typical projectile values, the energy density will start quite low as well.
Additional problem comes from the ability of vortex to impart the energy to the target. This depends on relative hardness and toughness of the projectile and the target with toughness enhanced by the relative densities of the colliding objects. Your vortex is made of air, which is neither hard nor tough and has much lower density than the targets it hits. Most of impact energy would not be transferred to the target, I think, and would instead create strong turbulence around the target.
This might be okay if the air was unable to flow around the target, but when hitting a target like a human the air is able to flow around and much of the kinetic energy will simply keep going as a chaotic vortex system behind the human.
A weapon for knocking people down might be believable.