Why do my bass drivers shake violently listening to vinyl

Different signal from each. Vinyl putting out more bass than digital. The question is how does it sound? Is it distorted? If so, then you may have a problem. 

It sounds incredible! I'm just concerned that I may be doing damage to my speakers

Rarely does a problem allow the sound to be good, much less incredible. What kind of music do you notice this with?
Some music causes this phenomenon of very active woofers. Heavy metal and hard rock is one of them . Plus your speakers have a a " 8” Carbon-Loaded Cellulose Flat-Cone Woofer with Ultra-Long-Throw Motor Assembly" according to their literature (specs) I doubt you have a problem. unless the woofer moves without being fed any signal

8” Carbon-Loaded Cellulose Flat-Cone Woofer with Ultra-Long-Throw Motor Assembly

I copied this from the Vandersteen site . Notice " Ultra-Long-Throw" IMO, that is what you are experiencing, an ultra long throw of the wooferThat said, @erik_squires may be right about a subsonic filter. It can’t hurt but will it help? I’d take a video and send it to Vandy tech service. They should know if this is normal for that speaker
You might also do as others have suggested by pulling the TT out of the dormer. Bass may be feeding back thru the cart and thus exacerbating the bass issue

@lewm

I generally prefer to go from the simple to the complex, in trying to solve any problem. Simple is to either move your turntable in your listening room or suspend it, or both. Complex is to add a subsonic filter

I agree wholeheartedly with this approach

@erik_squires

I have no real knowledge of electronics as you do. But I have been involved in using stereo systems for 60 yrs and can only speak from my observations. That said, I have seen my subwoofers pump wildly...or at least what I thought was wildly. It was much more than 10 mm. But I have never had any sound issues nor break downs. That said, Why does it still sound " incredible " according to the OP? I know what sub sonic means but I have heard it before in a system..at least thats what I thought it was. It DID go away with the subsonic filter
BTW I'm seeking answers not a fight (-:

@audionoobie
I kindly ask that you post your solution after you get this problem solved. I'm just curious