"In some cases racism is a prominent, or even the chief theme, while in other works critics have revealed racist attitudes that serve as underlying assumptions, but may not be immediately evident to the reader."
In An Occurrence At Owl Creek Bridge, there is no overt racism in the story, the prisoner is white and so are his executioners. But it is an underlying theme because the plot involves the conflict between the North and the South, the Civil War. Peyton Farquhar has risked his life to help the South gain an advantage in the war.
Although he is not a soldier, he longs to perform some task that will bring honor and recognition to his name. Therefore, it is easy for the Federal (Northern) Scout, undercover spy, to interest him in the plot to blow up Owl Creek Bridge.
Even though Farquhar knows that if he is caught, he will be executed, he agrees to carry out the mission.
The actions of Farquhar, who is an Alabama plantation owner, a supporter of slavery, has risked his life to help the South maintain the status quo, the institution of slavery. Therefore, the underlying racism lies in the motivation of the main character's actions.
His desire to help the South win the Civil War is based on his need to maintain the plantation society, his whole life, is rooted in racism.
Check out the line in Part II when Mrs. Farquhar gets the soldier a drink of water with her own white hands.
I have taught this story many time in American Literature and have never included a racial element in the teaching. The only fact that could possible be construed as racist is Farquhar's alliance with the Confererate cause during the Civil War. I would be hard pressed to find any real racial undertones in this story besides the fact protagonist has a vested interest in the outcome of the war because he is a southern planter.