The primary, driving conflict of "The Sniper" is man vs. man, given that it follows the lethal confrontation between two snipers amid the Irish Civil War. The Republican sniper is the story's protagonist while his brother (though he will not realize this until the story's ending) is its antagonist, with each intent on killing the other.
However, stories are rarely limited to only one form of conflict, and "The Sniper" is no different. In addition to man vs. man, it also contains internalized conflict, especially after the sniper is shot. Wounded and left with only one functioning arm, his situation requires mastering his own battered body and maintaining his nerves while facing the highest of stakes, stuck in a literal life and death struggle.
Finally, I think there is a strong argument that this story contains man vs. society as well, given its setting amid Civil War conditions. Aside from the lethal contest between brother and brother, for example, we also observe an old woman acting as an informant (who is gunned down on the street by the sniper). Indeed, the Sniper himself seems to recognize the brutality and insanity of these Civil War conditions, and he is shown viscerally reacting against it:
The sniper looked at his enemy falling and he shuddered. The lust of battle died in him. He became bitten by remorse. ... He began to gibber to himself, cursing the war, cursing himself, cursing everybody. (Liam O'Flaherty, "The Sniper")
Thus, while this story's primary conflict is man vs. man, following the lethal confrontation between the two enemy snipers, that driving conflict is deepened by these other forms of conflict (both internal and external) that unfold simultaneously to it.