In order to address the existence of a soul in Victor's creation, in Mary Shelley's novel Frankenstein, one must understand the different aspects of possessing a soul. (The following definitions of a soul have been taken from thefreedictionary.com.)
1. "The animating and vital principle in humans, credited with the faculties of thought, action, and emotion."
In this sense, the monster most certainly has a soul given his animation (life) and his ability to think, move, and emote. In chapter nine, the monster reveals his initial feelings and emotions.
It is with considerable difficulty that I remember the original era of my being; all the events of that period appear confused and indistinct. A strange multiplicity of sensations seized me, and I saw, felt, heard, and smelt, at the same time; and it was, indeed, a long time before I learned to distinguish between the operations of my various senses. By degrees, I remember, a stronger light pressed upon my nerves, so that I was obliged to shut my eyes. Darkness then came over me, and troubled me; but hardly had I felt this, when, by opening my eyes, as I now suppose, the light poured in upon me again. I walked, and, I believe, descended; but I presently found a great alteration in my sensations.
This passage allows readers to recognize that the monster has feelings, can think, and can move.
2. "A person's emotional or moral nature. "
Eventually, readers come to recognize that the monster is emotional and moral. While his actions may speak against this initially, the monster's final dialogue proves he possesses a soul.
“And do you dream?” said the daemon; “do you think that I was then dead to agony and remorse?—He,” he continued, pointing to the corpse, “he suffered not in the consummation of the deed—Oh! not the ten-thousandth portion of the anguish that was mine during the lingering detail of its execution. A frightful selfishness hurried me on, while my heart was poisoned with remorse."
Here, readers can see that the monster does feel true emotion and possess morals. The monster recognizes that his past actions were wrong and his asking of forgiveness ("Oh, Frankenstein! generous and self-devoted being! what does it avail that I now ask thee to pardon me?") proves his possession of a soul.