William Faulkner organizes the story in non-linear fashion because he is presenting the narrator's memories. The reader can infer that time has passed since the last incident—the townspeople's search of Miss Emily's home—that they mention, but there is no way to determine how long ago that occurred. In fact, because the entire tale is dependent on one person's perspective, the reader cannot assume that the narrator is reliable. The story has many qualities in common with folk tales or talk tales. We learn little more about the narrator beyond their sense of identification with the town's white population and their strong interest—one might even say obsession—with Miss Emily and her family.
The author had free rein in deciding on the voice to use and within his opting for first-person (who would be the narrator). The dominant feature of nosiness requires a narrator who seems to have access to a wide variety of local sources, such as a stereotypical "busybody." The narrator uses "we" to identify the people who toured the house, uninvited, after Emily died, but does not actually claim to have been an eyewitness to the sights mentioned.
A different story would have resulted if Faulkner had identified the narrator. Chronological sequence would not work well because of the long time span, unless he had opted for a third-person omniscient narrator.