I’ve built a vacuum enclosure out of some recycled wood I got from a demo at work. My idea is to have 3 compartments: one for the vacuum lined with rigid insulation, one for the cyclone, and an airtight drawer for the chips. A rough sketch is attached without dimensions. There will be a 2" wide vent down the back of the enclosure for the exhaust from the vacuum.
Thoughts, suggestions and constructive criticism are welcome. 
Mostly finished and assembled:
My only concern would be heat buildup in the vacuum itself. But that’s just a gut feel, with no practical experience or training in such matters.
Yea, the vac is going to get really hot in there. Might be alright, might kill itself after an hour.
If you really want to isolate the vac and its noise, put it in another room. With decent circulation.
If you’re basing the noise on your current vac only, I understand that adding the cyclone can make the vac much quieter.
It didn’t on mine.
What about running the sound baffle horizontally above the vac and having it exit “straight” out the back? Something like the inside base of https://www.youtube.com/watch?v=NKkbq1fcKz4, but at the top instead of the bottom. At least then the heat could continue to rise out by convection after the vac was turned off. I’d also consider adding some (baffled) inflow vents to the vac part of the cabinet for extra cooling air.
You may also want to consider carpet padding or some other “soft” insulation rather than rigid foam. You want the sound energy absorbed, not bounced back. Pressboard egg cartons will absorb some sound waves and scatter the rest pretty well. Don’t know how well the styrofoam ones do in comparison - haven’t tried those.
That’s why I said “can”… 
my guess is that if you can remove the bulk of the dust and chips before the vac, the less it has to work to maintain airflow. Again, not a guarantee, but a possibility.
Or your vac sucks… 
I thought the motor mostly cooled itself through the use of the air going through the vac. So I am not so worried about the heat, because it will constantly be moving new air through it.
It’s an option. It just means the box has to be taller and the air has to travel from the vacuum exhaust past the vacuum again to reach the outlet. I was thinking of adding a different material on top of the foam as well like carpet tile (also salvaged). So 5/8" ply box with rigid foam and rubber backed carpet tile inside that… Worth a try anyway.
This is where I got the initial idea from: http://www.startwoodworking.com/post/how-silence-your-shop-vac
My Shop Vac has a small fan on the top that comes on when the vacuum runs. You can feel the warm air being expelled from it. If the vacuum tried to use the air flowing through it, it could run into issues with dust and dirt contamination.
Point any openings into the box down. Then put a layer of carpet on the floor under the opening. The carpet will help absorb the sounds leaving the box. Don’t point the openings back into the wall as it will just bounce of back at you.
sounds good, and already shown that way on my drawing. 
Yea, if they cooled with the dirty air the motor wouldn’t last very long. 99% of the noise is the tiny bastard of an impeller. It’s a shame they don’t make them with larger slower moving impellers, they would be quieter.
These are all the reasons I’ve heard to never let the filter get clogged :). My shop vac does not have a fan on the top. My shop vac is also as old as me though…
Weird, I’ve never seen one like that before. You’re also not that old…
Shop vacs from quite a few years ago were cooled by the air flow though the filter. Since it increases the chance of the motor burning out from a clogged filter, modern vacs use a secondary fan on top that isn’t part of the vacuum air path for the motor. But it does affect the question of how best to enclose the vacuum.
Nice idea, like others tho I would be concerned about heat buildup. I run 3 vac type extractors in my furniture workshop and they do get warm. One in in an open front enclosure and it is quite warm after using it.
Is your vac still noticeably loud with the router on too?
Yes, it’s a lot louder than the router depending on speed.
While just about everyone has been bringing up the heat issue (and it is pretty valid) my question is what is your typical use case with the vacuum?
If you are sanding many square feet of board from 80 to 400 grit you may run into heat issues due to long periods of time with the vac on.
If you are cleaning up after you send chips everywhere from your router that doesn’t have good chip collection (like my current router) then you probably don’t need to worry about heat as you won’t have it on for more than a few minutes.
