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. 

There was the Acoustic Precission Eikos FR-1 produced some time in the 90s and the result of a collaboration between Tom Evans, Patrick Handscombe (presumably not the cricketer) and Ted Jordan in the UK. It was a single driver in a polystyrene foam box with a thin skin on the outside. It worked very well when I heard it at Tom’s place in Wales and my local hifi shop carried it for a while but I don’t think it sold that well. People prefer something more substantial for their money it seems. The finish wasn’t very robust I’ve heard.