I grew up there and my parents still live there, and I am not aware of a public range either.
At one time years ago anyone could go by the Sheriff's office and get the key to the Sherriff's range, leaving only your contact information and the promise to have the key back later that same day. I took advantage of this twice, and I knew at the time it was too good to last. The Sherriff's range is a really nice facility and most people can't handle having access to something nice... It was not available when I checked once during the last administration. Now that Chuck Mayfield is in office, there may have been some changes or there may not. I don't know, but it might be worth checking. I am sure it wasn't a priority as he has had his hands full, but from what I understand he has been doing a great job in office.
The only other range I am aware of in Natchez is where the Cowboy Action guys shoot and it is private, requiring a member to be present with any guests.
I believe a range was opened across the river just west of Vidalia, LA, a few years ago. I haven't heard much about it though so I don't know if it is still around. Concordia Pawn & Gun might have some info about it. I think their number is still 318-336-8925.