Bonnar signed for Celtic from Arbroath on 9 August 1948 and made his league debut in the 4–0 victory over Clyde on 6 November that year. Despite playing second (and even third fiddle) he was given his back chance on Hogmany 1949 when Willie Miller was injured and it was Bonnar and not Alex Devanny given the nod to play. He was soon heralded as "Celtic's best since John Thomson". Although small, Bonnar was agile enough to make over a century of league appearances for Celtic, and over 200 appearances altogether. Bonnar was not particularly tall for a goalkeeper but was able to produce moments of magic. He was Celtic's goalkeeper and hero in the 1953 Coronation Cup final against Hibs, where he kept their "Famous Five" forward line at bay, it is considered to be one of the best performances ever by a Celtic goalkeeper. Bonnar achieved 49 clean sheets in his 180 competitive appearances for Celtic.