Frankenstein's monster did not begin with a soul, but he acquired one over time. A soul can be defined as self-awareness. In the beginning, Frankenstein's monster was not aware of himself and did not express emotion beyond whether something was good or bad. He was unaware of what he wanted beyond being warm, cold, hungry, thirsty, or tired. This is not self-awareness and simply survival. Over time, he begins to express more and more emotion, as can be expressed with his discovery of the books in the forest.

After time had past, Frankenstein's monster finds a copy of Paradise Lost. While reading it, he identifies with the characters of Adam and Satan. In order to identify with a character, one must have some idea of what you are like yourself and your emotions. Frankenstein's monster clearly expresses why he identifies with these characters based on his experiences and emotions. This requires the awareness to understand past experiences and past and present emotions and therefore, self-awareness. This means that Frankenstein's monster does, at least by my definition, have a soul. It is unlikely that Frankenstein's monster developed a soul at that exact point as he expressed emotion and thought before, but this example is a good indication of him having one.
Beyond that, he demonstrates the capability to care for others. This is also a good indication for a soul. This is because to care for others, a person must recognize that the other person is in a bad situation and that they themselves would not want to be in a similar situation and would want someone to help them. Frankenstein's monster stops taking food from his poor neighbours when he realizes that this hurts them and starts finding food for himself, a harder pursuit. He additionally starts supplying them with wood. He does not benefit from this, except to make the other people happy. This is caring for others when it is not necessary and shows that he is simply doing these things because they make others happy, which makes him happy. This expresses emotions and empathy, both of which require some level of self-awareness. Therefore, he has a soul even that early on.