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He joined famous nursery club Belvedere FC at the age of 8 and played for them for 9 years until he signed for Stoke City as an apprentice in 1991. He signed professional forms the following year but never made the breakthrough to the first team and returned to Ireland to sign for Sligo Rovers in 1994. He lost 2 FAI Cup semi finals and a FAI League Cup during his time at Sligo all to Shelbourne where he made 60 league appearances.In the 94-95 season he was voted Sligo player of the year was selected for the Bord Gais All Stars selection as well as being nominated for the PFAI Young Player of the Year award finishing runner up to Billy Woods of Cork City. When his contract expired at the end of the 96/97 season, Robbie moved to Coleraine. Coleraine won their first 7 league games but were pipped to the league title by Crusaders they did win the Ulster Cup defeating Crusaders in a penalty shoot out with Brunton scoring in the shoot out and also scoring the og which brought Crusaders level. After only one season there, he was on the move again - this time to then champions Derry City. Derry failed to regain their title in a disappointing season with Brunton missing 10 weeks with a serious knee injury and with a new manager coming in, Brunton was surplus to requirements. He signed for Bohemians and made his league debut on 28 August 1998 against Finn Harps. He spent 2 seasons at Bohs winning a runners-up medal in the FAI Cup again losing to Shelbourne before he moved to Buckley Park and Kilkenny City. The tough tackling left back - he was sent off a total of 13 times - has since played for Monaghan United Dundalk, Newry Town and Cliftonville.
- "Leeman agrees Glens deal". BBC Sport. 28 January 2004. Retrieved 4 April 2011.