Following youth experiences at Invergordon, Gunn began his career with Scottish team Aberdeen in 1980. The presence of Scottish international keeper Jim Leighton meant that Gunn only played 21 games before he signed for Norwich City in October 1986.

Gunn did, however, receive Scottish League Cup and European Cup Winners Cup winner's medals while with Aberdeen. At both finals, he was an unused substitute. Gunn forged a good relationship with then Aberdeen manager Alex Ferguson, and in 1997 Ferguson brought Manchester United to Carrow Road for Gunn's testimonial. While an apprentice at Aberdeen, Gunn had been a babysitter for Ferguson's children.

Gunn then spent the majority of his playing career at Norwich, becoming one of the most famous goalkeepers in the club's history. Gunn initially looked set to join Norwich in the summer of 1986 after they had sold Chris Woods to Rangers for £600,000, however an injury to Leighton meant that Ferguson kept Gunn at Pittodrie until October.

He went on to make 477 first team appearances for Norwich. He was twice voted Norwich City player of the year in 1988 and 1993 and during the latter season, Norwich finished third in the Premiership, their highest-ever finish.

Gunn retired following a short spell north of the border at Hibernian.

Gunn made six appearances for Scotland, four of which were in World Cup qualifiers in 1993, when his career was at its peak.

In February 2007, Gunn moved back into Football becoming 'Club liason' for the Canaries.