As james014Aura said, both the Bellflower Network and Firebrands would be the best choices for anti-slavery organizations. The LOCG has a bunch of material for character options for the Firebrands and if you can preview the content the 3rd book of the Age of Ashes adventure path has character options for the Bellflowers.
I'm sure there are some Desna followers in the region, there are always some Desna followers somewhere everywhere.
The wiki says Liberators are welcomed by Cayden and Cayden is worshiped in the shackles so I can see at least a small group there to at least deal with some of the slave trade.
The Gray Corsairs are all about stopping naval slave traffic so I can see a large portion of them in the Shackles.
Steel Falcons work around the region (specially outside Andoran) freeing slaves and causing revolutions. So a small group along with some Twilight Talons could very well infiltrate some pirates to free slaves and cause mutinies.