You can build a “sound proof” enclosure for your shop vac. I haven’t done it yet, but I have considered it.
Key points I’ve taken away from my research is that in order to significantly reduce sound, you need a lot of mass, mechanical separation and layers of different density/thickness. For instance, you could build a box out of 3/4″ MDF or some other heavy dense material, then line the box with another material, like sheet rock or 1/2″ plywood. The key is to mechanically separate the two layers and the way to do that is with a material like “Green Glue” (I’ve also read that Roberts 3095 carpet adhesive works really well at a fraction of the cost of Green Glue). You could then line the inner box with a mass loaded vinyl material.
Another key point is that acoustic foam (egg crate foam, etc.) is not really meant for sound proofing. Acoustic foam is hung in recording studios and whisper booths to prevent echos, not to sound proof them.
I found this blog post helpful/interesting: https://acousticalsolutions.com/soundproofing-small-loud-widgets/