I don't know about your GM but on my van and my buddy's truck, when you remove the interior door panel, there's three bolts holding the mirror to the door. There may be another trim piece over the bolts as well - the ones I've seen require you to remove the panel before you can remove the smaller piece of fascia.
You can check the tightness on those bolts and snug them up and that may help the rattling.
You have to be careful, because the mirror housing that the bolts screw into may be plastic and you may strip the bolt right out if you tighten them too much.
That said, I have an aftermarket mirror on my van, and it shakes even after snugging up the bolts. So there may be an issue with the mirror glass mount in the mirror housing itself.