There are some clues in the story to give us some ideas about when this story takes place. In the first section, we are told that Miss Emily was relieved of the obligation to pay taxes in 1894, going back to the date of her father's death and "on into perpetuity (288). We do not know the date of her father's death, but we do know that the next generation sends a delegation to investigate when her taxes remain unpaid. In Section II, we are told that she gets rid of the delegation, as she had gotten rid of the delegation sent thirty years earlier, two years after her father died. So we can make a guess that would take the action up to 1924. We are also told in Section V that when Miss Emily dies, there is a room that must be opened, a room that no one has seen for more than forty years.
There are other time "clues" in the story, for example, that she died at age 74 and that she gave china-painting lessons for six or seven years when she was forty. If you read the story carefully, you should be able to make an educated guess about the entire period in which the action takes place.