This is an interesting question. A story does not necessarily have to have an antagonist, but it could be argued that the antagonist in An American Tragedy is poor Roberta Alden, the girl who loves Clyde and whom he loved before he met the rich and fascinating Sondra Finchley. Roberta becomes pregnant and insists on having Clyde either procure an abortion for her or else marry her. Clyde no longer cares for Roberta because he sees an once-in-a-lifetime opportunity to move into the upper class by marrying Sondra. But he cannot just abandon Roberta. She would expose him and ruin his precarious relationship with Sondra and her whole glamorous world. But if he married her he would also lose his relationship with Sondra and the privileged world she represents. The situation is somewhat similar to that dramatized in Ernest Hemingway's short story "Hills Like White Elephants." There the American is the protagonist and the pregnant girl he calls Jig is the antagonist. The American evidently feels that by assuming the responsibilities of marriage and fatherhood he will be losing his dreams.
American Tragedy is a novel where the conflict is man vs. environment or, alternately, man vs. himself. The antagonist, then, would be not another person, but a situation, or a trait within the character himself that the protagonist must battle against.
If we accept that the conflict is man vs. environment, then the antagonist is the combination of social and hereditary constraints into which Clyde is born. Clyde is born in poverty to a family that is not especially resourceful or notable. Though he longs for the American dream of success, he cannot escape his background to attain it.
If we look at the conflict as man vs. self, then the antagonist is Clyde's overwhelming desire to achieve material well-being and success, and the moral weakness that allows him to pursue his desire at all costs. He is so consumed with this craving that he is willing to commit murder in order to get what he wants.
In either case, consistent with the naturalistic philosophy underlying the novel, Clyde the protagonist is ruined by the antagonistic forces he faces.