My quest for very low durometer silicone (in tubes, like regular gasket silicones) is in part for exactly that. It needs to be soft enough for a pea-sized pebble to deform almost to a disk when pressed between two fingers.
The silicone can form the kinds of soft rubber pads that you see many PC case fans with nowadays, but you can also "plug" the fan mounting holes. Then, one can use e.g. long (35-40mm) M3 screws with large washers (again with a thick coat of soft silicone rubber), so the only thing the fan contacts is the rubber. Even then, using a foam pad between the mounting surface and the fan helps. The silicone only ensures the vibration does not conduct through the screws.
If you drill the holes even larger, and for VESA100 mounts 3.5mm inboard (at exactly 100mm center-to-center distance), you can use foam cylinders and foam pads instead of silicone.
Note that for each screw, you'll want 3 washers and nuts: two surrounding the mounting plate, ensuring the screw is perpendicular to the mounting surface –– and here the silicone is much better than foam. The third washer and nut is on the outside, and here foam suffices.
The simplest option is to get sheets of foam, one exactly 25mm thick where you cut the hole for the fan. Gel-like cyanoacrylate glues have worked well for me, and you don't need much; center and corners has sufficed for me. (I tried to cut the corners so that the fan would stay put even without glues, but didn't have the tools: scalpel deforms the foam, and hot-wire cutter needs thick enough wire to hold in a rigid L or U shape.)
Then, sandwich the fan in 5mm to 15mm thick sheets, depending on how much depth you can give to the fans. These only have the round center hole in them for the air. On the outer end you need a semi-rigid surface (rubber sheet is rigid enough) for the screw nuts to push against.
Additionally, against any rigid surface, you can use rubber-backed adhesive sheets with foam on top like SilverStone SST-SF01 and similar. Open-cell foam has a lot of friction, and two sheets of foam will not generally slide at all. The rubber is good for dampening vibrations on stamped sheet metal, and adhesive backing keeps it in place. This way, the screws have almost no pressure at all, they only keep the assembly in place against gravity. The second washer and nut can then be plain, crushing the foam between the rubber sheet and mounting surface. The circle you cut out is still useful for the washers and nuts on the other side of the mounting surface.
If you don't mind the look, nylon locknuts and M3 threaded rod makes this quite easy to experiment with. My local hardware store sells these, and they're cheap. In the US, consider #4-40 threaded rod: it is thinner, but that's okay, as the weight of the fan and the foam is easily borne by even smaller. I'd use M2 if that was more easily available, although it starts to bend a bit too easily at 40mm+ lengths.
For VESA mounting both the fan and the stand, I'd use thicker bolts, maybe M4 or M5, depending on the total device weight. Another option is making a bent steel bracket with two VESA100 mounts side-by side (one on top of the other); the bends giving ~ 1-2mm sheet metal the required stiffness. That way, the fan intake is not blocked by the stand mount. (This is wrt. Rigol DHO800/900 series oscilloscopes and similar.)