I have stated elsewhere my view that in a story of this type the protagonist and antagonist are great powers whose conflict affects the fates of unimportant individuals like Peyton Farquhar (who seems modeled after John Wilkes Booth). The protagonist and antagonist in my view would be the North and the South. Since the North seems to have the momentum at this point in the Civil War, I would choose the North as the protagonist and the South as the antagonist. The Union officers and soldiers do not seem to be acting as individuals but as pawns being moved by some invisible force above and beyond them. Even Peyton Farquhar seems to be moved by something he only vaguely understands. Hundreds of thousands of men fought and killed each other without hating each other or really understanding why they were fighting. It seems to me that this was the feeling that Ambrose Bierce--famous as a cynic, pessimist and nihilist--was trying to capture in his story.
The human antagonist(s) in Ambrose Bierce's "An Occurrence at Owl Creek Bridge" would have to be the Union soldiers who are about to hang the protagonist, Peyton Farquar. The Union soldiers are Farquar's direct enemies, and with the lack of other characters in the story, there can be little debate on this question. (The Union spy who tricked Farquar could also be considered a minor antagonist.) It could also be argued that the hangman's noose serves as an antagonist: It is the rope that Farquar hopes to somehow evade as he stands upon the railroad trestle awaiting his fate. And it is the rope that eventually serves as the instrument of his death.