There is no direct conflict in the story, but Walter Mitty is fighting back subconsciously against the conventions of society and his role in it. Society has made him a timid and quiet man without any larger ambitions, and he is pushed around by pretty much everyone in his life. However, in his daydreams, he is a man of action, a hero who saves the day and is admired by everyone he meets.
"The Old Man'll get us through," they said to one another. "The Old Man ain't afraid of hell!" . . .
"Not so fast! You're driving too fast!" said Mrs. Mitty. "What are you driving so fast for?"
(Thurber, "The Secret Life of Walter Mitty," bnrg.cs.berkeley.edu)
In that sense, the central conflict is Man versus Society. Mitty dreams of adventure, but his only real drama is forgetting the errands his wife wants him to run while she is at the salon. He doesn't run into any actual adversity, but he dreams of conquering vast difficulties and coming through with his dignity intact. Society, however, has other plans, and he ends the story much as he starts it, in a daydream with no other future ahead of him.