What is the moral lesson of the short story?

I don't think there is only a single moral in this story.  However, I think a major moral lesson of this story is focused on how war affects people.
"The Sniper" does a great job of showing the dehumanizing effects of war. Regardless of the specific conflict, war has a way of driving all humanity out of the people fighting in it and living with it. People are no longer people; they are simply tools that fight, or they are targets to be eliminated. The story helps tell war's impersonal nature by keeping all of its characters impersonal; no names are used. There is a sniper, another sniper, an old woman, and so on. I think readers are lucky to have that much information because to the story's protagonist, who or what a person is doesn't matter. To the sniper, everybody is a target of equal opportunity. It doesn't matter if the person is young, old, man, or woman. They are targets to the protagonist. He doesn't see them as human people.
The shock of the story comes at the end when readers learn with the sniper that he has shot and killed his own brother. Based on all of this, I think that a moral of the story is to be aware of how war affects the men and women who fight it. They form a desensitized, impersonal detachment from humanity that might cause them to do something that they will regret forever.