I recently acquired a brand new rubber bumper strip for the rear
chrome-style bumper. Part number 251 807 421. From the rubber side, it is
like very other one I have ever seen. From the backside, the holes for
mounting the attachment buttons are not the usual slots that allow you to
snap in the plastic buttons, but are double horizontal slots at each
attachment location.

The two recesses for the bumper attachment bolts are in their usual places.

Does anybody know what kind of mounting buttons this part needs, and where
to get them?

It's a VW-AUDI part, if the imprint is to be believed.

Thanks,

Jim