I agree that the climax--the point of highest interest and suspense--is the moment the townspeople herd into Miss Emily's house after years of only her manservant having access. They finally open the door to her most private place--her bedroom--only to discover that she had been sleeping with the Yankee they all condemned her for seeing in public. A southern lady just didn't do these sorts of things, but then again, no one made the connection that he was up there as they were sneaking into her yard to spread lime in order to quell the stench of the "mouse" or other animal that had died somewhere on Miss Emily's property. It just isn't good manners to tell a southern lady she smells...and she hid behind the rules of gentile southern behavior and escaped the crime of murder all the while protecting her good reputation rather than allowing Homer to have his way with her and then publically humiliate her. Who said the south lost? In Miss Emily's world, she (the South) was more than victorious.
After Emily is buried, some of the people of the town force open the room above the stairs in her house. They find the skeletal remains of Homer Barron, but the climactic part of this is when they find the long strand of gray hair on the pillow beside Homer. The narrator hints several times in the story that the town suspects Emily of killing Homer, but I don't think any of them dreamed that his body had been kept in Emily's house for over 40 years. The only hint of this is when she wouldn't allow her father's body to be removed for three days. But at that point, the reader tends to assume it's because her father was all she had.