Why do my bass drivers shake violently listening to vinyl

Erik is 100% right. Back in the day, rumble filters were a feature on many receivers as it was often needed for the very problem you are experiencing. I had the issue myself sometimes.

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