I like this question a lot. In a novel about the Civil War and post–Civil War rebuilding, a reader would assume that William's courage must certainly be focused on heroic acts of violence and action. That isn't the case in Shades of Gray.
One of William's first courageous acts is simply being able to attempt to move on from all of the deaths of his family members. His father and brother were killed in the war. His sisters died of disease. His mother died from grief. William has every right to "throw in the towel" and quit on everybody. He has every right to be angry and resentful toward life. He is an angry character at the start, too, but it is directed at the "northern Yankees." Now an orphan, he agrees to live with his aunt and uncle in Virginia. William is not thrilled about the arrangement, because he believes his Uncle Jed is a coward.
This is one way that William shows courage. He is moving to an unknown location, to live with family he doesn't know that well, to be with an uncle he doesn't respect, but William musters up the attitude to make the best of it. He is respectful to all of them and even shakes Jed's hand when meeting him. William works hard on the farm to pull his own weight.
The rest of the book is about William's slow and gradual change in attitude toward Northern soldiers, Southern soldiers, and his Uncle Jed. In a nut shell, William realizes that not all Northern soldiers are evil. Many of them are similar to people he knows in the South. William comes to learn that the Southern army is not as pure and wholesome as he once thought. Most importantly, though, William learns to respect his uncle. He sees that Uncle Jed is a hardworking man of principle. He cares for his family and his community. He doesn't support violence to solve problems, and that's why he didn't fight in the war. William may not agree with that, but he learns to appreciate that Uncle Jed acts on his principles.
William's complete 180 degree shift in opinion toward the beliefs he held firmly to the core of his being took great courage. It's not easy to admit that you were wrong. It's not easy to accept that what you once thought completely true is false. It takes courage to admit fault and work to better yourself because of it. William does this throughout the novel and that is why he displays courage.