Two suggestions I have tried:
1. Obviously, if you are worried about the noise, you should close the lid when you vaccuum. This alone will be a slight reduction.
2. Place the machine on top of piece of carpet. My machine placed on my bare shelf runs very loud. However, when I turned it on once while it was on the floor (carpet), it was drastically quieter. I recently picked up a new remnant, but haven't yet had the time to cut and use.
As mentioned above, the one concern is heat. Mine gets hot when I clean a few LPs (three separate vacuums per side). Placing the unit on top of the carpet may cause the temp to go up a bit. I'll still definitely do it, as I rarely clean more than 3-4 LPs at a time.
Heat is the reason I wouldn't add any sound absorbing materials to the inside of the machine. Try these easy fixes first.
-d