Does this story relate to William Faulkner's life during the period in which he wrote it?
William Faulkner's story "A Rose for Emily" was first published in April, 1930. Emily's "timeline" begins with her birth during the Civil War (1861-1865) until the early part of the 1930's.
Faulkner is born in 1897, placing his birth at merely twenty years after the southern Reformation ends in 1877. This means that Faulkner's immediate family is part of that reformation time where the South is searching for a new sense of purpose and self-identity.
This is why we find most Faulkner's biographers agreeing in Faulkner's deep interest in Southern history from a very early age, and especially in taking part of the daily lives of the elders of his town in Mississippi, who would sit around the town square to reminisce in times gone by. This is evident in his biography William Faulkner: His Life and Work
...there he sat or stood motionless, quiet, as though held fast by some inner scene or some inner sense of himself
Basically, Faulkner acquires his information from primary sources and, for that reason, he has a qualitative resource that can provide first-hand accounts of what actually happened during the Civil War, during the years before, during, and after slavery and, especially, during the reformation.
This being said, you can certainly see "A Rose for Emily" relating events that Faulkner did not experience first-hand; however, his family, his elder neighbors, and the immediate community was certainly there. So powerful are his sources, that we even question whether Faulkner was actually alive during Emily's "life", and whether he is relating the story of a real woman- one that an entire town, as well as Faulkner, himself, knows personally.
At first we were glad that Miss Emily would have an interest, because the ladies all said, “Of course a Grierson would not think seriously of a Northerner, a day laborer.” [...] They just said, “Poor Emily. Her kinsfolk should come to her.” She had some kin in Alabama; but years ago her father had fallen out with them over the estate of old lady Wyatt, the crazy woman, and there was no communication between the two families. They had not even been represented at the funeral
Furthermore, Faulkner even uses their voices to relate Emily's story. We can definitely agree that what Faulkner tells us about Jefferson County are the stories, experiences, and events that were once related to him from verifiable sources since he was a child.
William Faulkner was born, raised, and spent a good part of his adult life in or near small, rural Mississippi towns much like the town of Jefferson, the setting of "A Rose for Emily." As many of Faulkner's biographers have noted,
Mississippi marked his sense of humor, his sense of the tragic position of Black and White Americans, his characterization of Southern characters, and his timeless themes, including ﬁercely intelligent people dwelling behind the façades of good old boys and simpletons. (See cite below)
Although the central theme of the story—a woman's life originally blighted by her father's selfishness and the tragic consequences—is not necessarily Southern, the setting of Jefferson, the nature of the narrator (a collective "southern" voice), Emily's adherence to southern cultural norms (i. e., choosing to care for her father), and Emily's position as the remnant of Pre-Civil-War southern aristocracy—all of these reflect life in the rural Post-Reconstruction South. These elements may not relate specifically to Faulkner's life, but they reflect the world he knew intimately.
Miss Emily's life in Jefferson relates to Faulkner's life in the sense that the world in which Miss Emily lives is precisely the world in which Faulkner lived most of his life. In fact, Faulkner knew this world so well that most, if not all, of his novels and short stories are set in a fictitious Mississippi county called Yoknapatawpha County, and several of his novels and short stories have Jefferson and its surrounding area as their setting. His creation of the world within Yoknapatawpha County has been called "one of the most monumental fictional creations in the history of literature." For writers like Faulkner and Flannery O'Connor, it is not possible to separate certain aspects of their "southerness" from their fiction.
In sum, then, we can say that although the main theme of "A Rose for Emily" may not relate directly to Faulkner's life, the frame of the story is constructed from Faulkner's understanding and love of the life and people of rural Mississippi.