The protagonist of The Sniper is, appropriately, the Sniper.
The Sniper is not given a name, which helps to establish that his identity is "a sniper", and not really an individual with a past. Much of the human element in the story revolves around the fact that this is a civil war, with personal lives and military obligations closely entangled; the Sniper does much to stifle his humanity, but it sneaks through from time to time. Despite his zeal and experience, he is obviously not a professional, at least in part because he takes unnecessary and emotional risks. His edifice momentarily collapses after he kills the Enemy Sniper, and we can imagine that it suffered immensely when he discovered the man's identity.
The Sniper is the protagonist because the story is told from his point of view, and no other character (and there aren't many of them) can be said to be the main personality of the story.