Why do my bass drivers shake violently listening to vinyl

Before purchasing a rumble filter, I would first see whether changing the location of your turntable in relation to the speakers has any effect on the phenomenon you are observing. Also, the addition of some isolation to the turntable base might help the problem. You might try to suspend the Technics turntable in someway. There are “feet” available for that. I would put all this ahead of adding a rumble filter in the signal path, only because the rumble filter is not without its sonic penalties. There is no free lunch in Audio. The only other thing I would add to the discussion is that someone mentioned DC voltage as the possible culprit. That is impossible since DC by definition has no frequency. Significant DC voltage delivered to the speaker voice coil could damage it, but it is not causing the piston like movements you are observing.

Per your system pics you now seem to have the TT placed in a dormer off to the side of the left speaker.

If possible try pulling the rack out of of the dormer, just temporarily, to see if the LF feedback lessens.

DeKay

All phono stages should have a rumble filter to cut off frequencies below 20hz. The result is cleaner bass and less strain on the amplifier and speakers!

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

I copied this from the Vandersteen site . Notice " Ultra-Long-Throw" IMO, that is what you are experiencing, an ultra long throw of the woofer

Your opinion in this case is not actually correct. :)

The phrase "ultra-long-throw" or "long throw" have no specific meaning, but are understood to mean that the driver MAY travel further without distortion than conventional drivers of the same size. Typically, larger drivers have longer throws.  This is specified by manufacturers as Xmax, and is usually in millimeters.  So you'd read it something like this:

Xmax : 10mm (the distance the driver can travel from rest is 10mm before distortion sets in).

That doesn't mean they flap back and forth more than any other driver of the same size and output.  The symptoms from the OP are definitely being caused by excess, and inaudible, bass in the signal which could probably be replicated by jumping on the floor. :)

You don't need an "ultra-long-throw" woofer to flap back and forth, just too much rumble in a ported enclosure.