Velka, Goddess of Sin does not seem to possess a Lord Soul or shard of a Lord Soul as there is no mention of this according to any lore sources.
She is described as a rogue deity and a black-haired witch with no allegiance save the hunting down of sinners, which seems to include the gods themselves. It is theorised that she is using the Chosen Undead to punish the gods; to attack them relentlessly until he succeeds, not even letting death stop him.
The Lords fear her, and everything related to her is locked away in the Painted World of Ariamis.
In Japanese mythology, crows are seen as servants of the gods and are often sent out for divine intervention in the world of mortals. In this vein, it can be surmised that the giant crow that transports the Chosen Undead to Lordran is a servant of some god. And, given Velka's obvious connection to crows it is safe to say that the giant crow serves or is her.
Design Works Interview
An interesting point, according to the Design Works interview with director Hidetaka Miyazaki;
The crow demons came about during the initial concept stages.
I always thought of the painted world as somewhere where things go to escape, and the bird men but no different. They were originally designed as worshippers of the Goddess Velka whose bodies were warped by their devotion. I think this obsession makes them really interesting characters.
It is important to note that Velka's Talisman, unlike other talismans associated with the gods, allow the user to use miracles using intelligence instead of faith.