This is an interesting question, because I think he does have a soul. If the soul is the life principle, then he must have one. The question is probably what it "looks" like. The creature (probably better than "monster") begins life with nobel characteristics, and is merely seeking to be loved. He exhibits all the characteristics that I associate with having a soul. His later actions, the result of his being unable to find love in the company of those we might expect to have souls, he beings the rampage that leads to killing of so many people. But killing does not mean he doesn't have a soul; after all, do we ever doubt that the murderers among us have souls? Or do we suspect that the tendency in them for evil has overcome the effort to do good?
And does he have any less soul than Victor who brings him into the world and then turns him lose without loving him?
I don't think so. I would suggest that he does have a soul, at least as I understand it.
The nature of evil that is expressed through the monster's behavior, systematically killing those people that Victor loves suggests that he has no soul. To be an individual without a soul, in my view, amounts to having no compassion for the pain and suffering of others. To feel that to punish someone else is a victory for your own cause, your own pain, your own deficit.
The monster is not just killing, he is killing with the intention to punish Victor, an intentional, premeditated act, which is evil and soulless behavior.
Even though the monster has been created from the parts of human beings, he is not fully human. I would argue that as an abused and neglected person, the monster knows what he is missing, by not having love in his life or else why would he want Victor to make him a female companion.
His deprivation, his empty lonely life leads him to behave monstrously, like some human beings who feel that the world has cheated them, therefore they must lash out and kill or destroy the lives and happiness of others.
These people would be serial killers, who do not have souls like normal people. They have no conscience, which is where the soul resides. No one with a conscience would be able to murder the loved ones of anyone with the thought that he had a right to cause misery and pain.
The Serial killer who chooses his victims at random to satisfy his lust for murder, believes that his need outweighs the needs of his victims. They must sacrifice their lives in the service of his fetish, habit or obsession.
And finally, the soul is bartered for daily between the forces of good, God if you are a believer and evil, the Devil, again if you are a believer. Some people stay on the side of good, others choose evil, or are seduced by evil, or fall into evil because of their own stupidity, nonetheless, the forces of evil grow everyday.
The legions of those who will kill, maim, hurt and abuse, those who have no conscience, who cannot hear the voice of their own soul, who deny the essence of a soul in their victims, are evidenced in our society in our newspapers, news reports and all over the Internet every day.
For me, the presence of evil is easy to see, finding the presence of good, or the presence of God, if you are a believer, seems to be getting harder and harder to find.