Dresden discusses who has souls and who doesn't several times in the context of soulgazes. We have on-page proof that Red Court vamps can't be soulgazed and that White Court vamps can be. I assume that Black Court cannot be as they're essentially dead, but can't find a direct reference in the texts I have digitally. Jade Court is mysterious and unknown.
Outside of vampires, Fae and Demons do not have souls. Werewolves and fallen-angel possessed humans do. Wolf-weres do not.
Bianca is Red Court and soulless
I ignored it, meeting Bianca’s gaze as I came level with her. Vampires
don’t have souls. She didn’t have to fear my gaze.
Butcher, Jim. Grave Peril (The Dresden Files, Book 3) (p. 309). Penguin Publishing Group. Kindle Edition.
Chalk up the blanket statement that Vampires don't have souls to an overgeneralization by an unreliable narrator, because White Court vamps do appear to have souls:
Thomas is White Court and ensouled
It was only the second time I’d ever seen a soulgaze happen to someone
else. There was an instant where both of them locked their eyes on
each other’s. Molly’s eyes widened suddenly, like a frightened doe’s,
and she jerked in a sharp breath. She stared at him with her chin
twisting to one side, as if she were trying—and failing—to look away.
Thomas went unnaturally still, and though his eyes also widened, it
reminded me more of a cat crouching down in anticipation, just before
pouncing on its prey. Molly’s back arched slightly and a soft moan
escaped her. Her eyes filled with tears. “God,” she said. “God. No.
No, you’re beautiful. God, you hurt so much, need so much. . . . Let
me help you. . . .” She fumbled for his hand.
…
“What just happened?” “You soulgazed Thomas. You both lost
perspective. He nearly ate you.”
Butcher, Jim. Changes (The Dresden Files, Book 12) (p. 258). Penguin Publishing Group. Kindle Edition.
Mavra is Black Court and unclear The state of her soul isn't explicitly stated, but she's dead. I can read this as either Harry avoiding her eyes to avoid a horrific soulgaze or to avoid vampiric powers.
Her dead man’s eyes looked down at me, and I had an abrupt
understanding of why Michael had called her “it.” Mavra wasn’t a woman
anymore. Whatever she was, she wasn’t a person. Not like I understood
people, in any case. Those eyes pulled at mine, pulled at me with a
kind of horrid fascination, the same sickly attraction that makes you
want to see what’s under the blanket in the morgue, to turn over the
dead animal and see the corruption beneath. I fought and kept my eyes
away from hers. “Come, wizard,” Mavra whispered, her face utterly
without expression. “Let us test one another, thou and I.” I
Butcher, Jim. The Dresden Files Collection 1-6 (The Dresden Files Box-Set) . Penguin Publishing Group. Kindle Edition.
Non Vampires
There were only two kinds of people whose eyes I could meet for more
than a second or two: the people who had already met my eyes in a
soulgaze were one kind; inhuman beings from the Nevernever were the
other.
Butcher, Jim. The Dresden Files Collection 1-6 (The Dresden Files Box-Set) . Penguin Publishing Group. Kindle Edition.
the fae have no souls to gaze
Butcher, Jim. The Dresden Files Collection 1-6 (The Dresden Files Box-Set) . Penguin Publishing Group. Kindle Edition.
Demons, however, are a different matter. Demons aren’t people. They
don’t have the problem of having a soul, and they don’t worry about
silly things like good and evil, or right and wrong.
Butcher, Jim. The Dresden Files Collection 1-6 (The Dresden Files Box-Set) . Penguin Publishing Group. Kindle Edition.