Talbott began his football career as a starting tackle and halfback from 1912 to 1914, for Yale University. He was consensus selection to the 1913 College Football All-America Team. In 1914, he was named captain of the Yale team. Bud led Yale to a 28–0 victory over Notre Dame, ending the Fighting Irish 27-game undefeated streak. He repeated with All-American honors in 1914, making several major newspaper first teams.
After graduation, he became one of the organizers of the Dayton Triangles professional football team. He coached the local team in 1916 and again from 1919 until 1921. From 1922 until 1923 he was head coach of the University of Dayton football team who had just changed their name from St. Mary's University.