No guns are allowed to be discharged on the Appalachian Trail which 40% of the Appalachian Trail is on Federal land.
Even if you have a gun permit to carry the gun the guns on the Appalachian Trail are prohibited and you could get into serious trouble if you sneak a gun in and then discharge it.
It's not worth the risk to carry a gun and you'll usually be pretty safe without the gun on the Appalachian Trail.
A lot of people get by just fine hiking the Appalachian Trail without the need for a gun so it's not recommended to carry a gun on the Appalachian Trail to reduce your risk of getting into trouble by the Federal Authorities.
Although just like anywhere the Appalachian Trail can even be dangerous but it's still a pretty safe journey to hike through the Appalachian Trail without the need for a gun.