Why do my bass drivers shake violently listening to vinyl

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.

Audionoobie, welcome to vinyl. There is a lot of low frequency rubbish on records in the form of warps and irregularities in the vinyl. It can be made worse if your cartridge and arm are not matched correctly. unfortunately the flapping woofers increase distortion and waste power. Other than making sure your cartridge is appropriately matched the only other way to deal with this is a steep subsonic filter. A two way crossovers used with subwoofers will stop the flapping in your main speakers but will pass it on to the subwoofers so a subsonic filter is still necessary for the best performance. I use a very steep 8th order filter at 18 Hz. It is digital which is the best type of filter to use for this purpose.