Miss Emily's crime goes undetected for a few reasons, most of which are related to the social strata of the time and place.
First, no one ever gets beyond the first floor, and after awhile, no one even goes into the house. We know this because in the first section of the story we are told,
...no visitor had passed through the door since she had ceased giving china-painting lessons eight or ten years earlier.
Second, the townspeople are accustomed to the difference in social status between Miss Emily's family and their own families. This makes them reluctant to call upon Miss Emily. Again, in the first section, first, when the tax collecting delegation comes to call, she does not even ask them to sit. In the second section, we are told,
A few of the ladies had the temerity to call, but were not received.
Many of the townspeople are people who would probably have not even been allowed in at the front door, and would have only been allowed to make deliveries at the rear of the house.
Third, because Miss Emily has chosen a construction foreman, who is not even from the South, no one suspects anything when he is no longer seen. If he had been from the town or a person of means, his disappearance would have been a source of concern, and Miss Emily might have been a "person of interest."
If you are wondering about why Miss Emily's "Negro" manservant knew nothing, or perhaps did nothing, it is probably because he was not permitted to enter Miss Emily's bedroom at all, because he was too frightened as a black man to say anything, or, less likely, because he was devoted to Miss Emily.
There are many other passages in the story that support the reasons I have listed. Why don't you try to see what you can find?