Nope. If that were true the motor would be lugging with a/c off too. The pulley and tensioner are in use all the time.
I consider 'a/c on' lugging pretty much normal on higher mileage 4 cylinder cars, older motors do not have the idle rollover torque new ones do. From what I've seen the IAC bumpup in speed really is only enough to cover pretty close to new motors. Same issue shows when fan on or power steering is hit on limit. The IAC bumpup occurs, just not enough of it. Software issue. Engine too small. Put same exact a/c system on a V-8 and motor powers right through it because more torque at idle.
Now if vibrating dash when a/c on, look to your passenger side motor mount.