Triad dealers install our InWall and InRoom subs in cabinets all the time, and most subwoofers will work fine in a cabinet. There are a few problems with this type of installation, though...
You'll have to deal with rattles and vibrations of the cabinet and it's contents.
The sub has to be front-firing with no back port.
Some companies offer the option (Triad is one) of ordering a sub with the amp outboard; not built into the sub. This makes adjustments easier to make.
Companies may offer a grill-delete option so you don't have to pay for something you don't use.
Again, most brands of subs will work, even through a door with grill cloth on it. I'm using one in a cabinet in my exercise room, as a matter of fact.