It's my understanding that the linguistic root is in "purse".
That would make sense; A rabbit fur or leather purse worn on the belt (like a Sporran) was a common enough item in England in the medieval period, and the similarity in both appearance and location would be easily sufficient for the slang terms Coney>Cunny>Cunt and Pussy to develop from there. I don't have any evidence for this though, and in the absence of evidence, it is just one of those plausible sounding linguistic just-so stories that has about a 90% chance of being completely wrong.
'Purse' itself comes from the Greek 'bursa', meaning leather, via Latin, which borrowed the Greek word for use as a name for for money pouches that were made from that leather. The word 'pussy' to mean 'rabbit' probably comes from this root; Whether the word 'pussy' to mean 'cunt' comes from 'purse' directly, or via 'rabbit' or perhaps more plausibly, by a pun on both meanings by analogy to coney>cunny>cunt is lost to the mists of time.
Personally, I am just enjoying the opportunity to use the word 'cunt' a lot, while still sounding erudite.