I am afraid your suggestion of using suction cups to hold the LP on the vertical will not worksorry I wasn't more clear . . . I sometimes think what I write is clear, but often leave out key points.
The reason I mentioned suction cups is they are readily available in a size that would just cover the label. I know they wouldn't hold via suction (most young boys figured this out, and I was one of them). They would need to be held in place by a clamp - either around the outside of the record, or a shaft through the center of them and the record with a fastener on the ends of the shaft (so the record ends up becoming a wheel on an axle).
Grimace - your notion of using rubber washers is the same idea I think, just different raw material. Depending on how you do it you'll need plate of some kind to clamp the washer . I did this on a freshman Engineering design project in the early '70s. I picked the suction cups because I didn't have to fabricate a hub, and could buy them at the local hardware store. It was primitive but worked perfectly, though my record cleaner was manual, no vacuum. I suspect you have the wherewithal to do a more elegant implementation.
George, sorry you had to go to all that trouble on your theory and implementation of suction cups ;-)