The person who actually asks the question "Who shot first?" is Mr. Beach the Minister. I could see how you get confused about this because Sam's father is present at the table while the story is being retold. The reason why all the men at the table are so worked up about Sam's retelling of the Battles of Lexington and Concord is because they are Loyalists. They are also fairly upset at the retelling of the story by Sam because he is a Patriot and uses unfriendly terms to describe the British. He refers to the King's army as "Lobsterbacks" which was a derogatory term for British soldiers during the American Revolution.
Ultimately, this is so important to Mr. Beach, Sam's father, and the other farmers because they believe this will lead to war in the Colonies and this is treason. Sam's father makes this point clear when he says "God meant man to obey . . . children to obey their fathers . . . men to obey their kings."