The main conflict in this story may not be the most obvious one, but it is past versus the present.
Miss Emily is part of the antebellum South. Her home "had once been white, decorated with cupolas and spires and scrolled balconies" but now it was "an eyesore among eyesores."
In the old days, people looked out for one another. When Miss Emily must accept charity, a story is concocted to keep her from being taxed. But, "(w)hen the next gerneration, with its more modern ideas became mayors and aldermen, this arrangement created some little dissatification."
Faulkner carefully let's us see that the past isn't all good, the present not all bad. The slavery issue is of course, wrong, and Miss Emily's family complicit in its continuation. It is a good thing for it to have ended.
The conflict in the present, now, is that people have comes so distanced from one another. Faulkner makes the whole town complicit in their ignoring of human need and kindness. For example, in Part IV: So the next day we all said, She will kill herself"; and we said it would be the best thing."
There are other conflicts in the story: Miss Emily vs. Homer, Miss Emily vs. her own past, loneliness and human need. But the past and the present, for me, is most compelling.