He began his career in the late 1970s with his local youth side, Keadue Rovers. When he was 16 he had several trials with Leicester City and kept goal for them in the FA Youth Cup. A year later he was spotted by Sean Fallon who was scouting in Ireland for Celtic, and invited to Glasgow for a trial. Bonner impressed and signed for Celtic on 14 May 1978, becoming Jock Stein’s last signing for the Parkhead club.
Bonner went on to play 642 times for Celtic, with 483 league appearances. In total with Celtic, he won four League Championship badges, three Scottish Cup winners’ medals and a League Cup winners’ medal. His last appearance for Celtic was winning the 1995 Scottish Cup final under Tommy Burns. After the Scottish Cup Final victory over Airdrie, Bonner took on the role of player-coach and finally left the club in 1998 to work as a coach alongside former team mate Tommy Burns.
With 642 appearances, Bonner is the most prolific goalkeeper in Celtic history, and their fifth-most prolific player of all time.