The climax of the story is when Walter Mitty stands up to his wife and, when she is haranguing him about putting on his overshoes, says, "I was thinking...Does it ever occur to you that I am sometimes thinking?” While this is a very small revolt, it is rare enough for him to ask for anything from his wife that it constitutes a change in his actions and the climax of the story. He generally gives his wife a passive answer to her nagging questions and remarks, and then he returns to his world of make-believe heroism. He never lets his wife know about the fictions he makes up, and he veers between the real world and the world of his imagination without connecting them at all. In this small but significant climax, he brings together his two worlds and lets his wife know that he has other things on his mind. It's unclear whether this small change in Mitty's behavior will bring about a full-scale rebellion, but it is a change nonetheless.
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You could argue that there is no real climax in this story as Mitty does not ever change or reach a turning point. For argument's sake, you could analyze his final daydream. He envisions facing a firing squad as a form of suicide. Is he metaphorically killing his former self in order to become more assertive and stand up to his wife (or act more responsibly so his wife won't nag him so much)? Or does he really dream of dying as a form of escape? In reality, his limited options include standing up to his wife, positively changing his behavior, and/or leaving his wife. He could also kill himself or kill his wife. But considering that over the course of the story he his weak and passive, it's unlikely that he suddenly decides to make a significant change and much more likely that he simply continues to be a doormat to his wife and the firing squad is just the next daydream and many more will follow.