An Entwicklungsroman is a novel that deals with the development and growth of a character from childhood to maturity. Even though this novel does not deal with Edna Pontellier's childhood for a significant amount of time, it does present her growth from relative innocence to emotional and sexual maturity, and so I would suggest that the novel does qualify as an Entwicklungsroman.
When Edna meets and falls in love with Robert Lebrun, she is simultaneously recognizing that her lack of love or passion for her husband, Leonce, is a deficiency. She used to think that her marriage was what it should be, and she is "forced to admit that she knew of none better" than Leonce when all the ladies declare that he is "the best husband in the world."
However, Edna longs for more independence than married women were permitted to have, expressed symbolically in her desire to learn to swim and her desire to be near or in the water. She wants to go where she wants when she wants, love who she wants, and so on; in short, she wants the freedom that men seem to experience.
As she gains greater knowledge of herself and the world, she eventually comes to realize that she simply cannot be both independent of society's norms and also be accepted by society. She cannot be truly free of society's rules as well as still included by the society that makes those rules. Edna becomes truly mature when she recognizes these truths, though they render her incredibly unhappy. As a result, she chooses to give up her life but not her "self." She has told her friend that to give up one's life is not the ultimate sacrifice, but to "give [her]self" would be, and so she makes the choice to keep her self and refuse to live in a society that will not grant her the freedom she desires.