Are there speakers with cloth/soft enclosures instead of wood?

Sounds like mush. At least the bed board ones I made did.
Good enough to cover a squeaky bed.

Um, well the enclosure serves two purposes, keeping the front and back waves apart as well as providing part of the suspension.
There's open baffle, and some speakers rely heavily on thick foam as kind of wave guides.  
Assuming you made a baffle of say, organic Egyptian cotton, it would just blow back and forth, and essentially the speaker would be in free air.

SpeakerLab tried a cloth speaker back in the 90's. Put a lot of money into it. Got hung out to dry.