Mulligan started his senior career playing for Bohemians F.C. in 1963. However after only two games he signed for Shamrock Rovers in December 1963 . He made his competitive debut in a 7-1 win over Bohemians on the 26th of January 1964 and scored his first goal for the Hoops on the 23rd of August 1964 in a 4-0 League of Ireland Shield win over his old club .
He won the FAI Cup in 1965, 1966, 1967 and 1969. During the 1963-64 season he was part of the side which won seven trophies. He represented Rovers in European competition 10 times. In the summer of 1967 he was part of the Rovers team that represented Boston in the United Soccer Association league. In 1968, they sent him on loan to the Boston Beacons of the North American Soccer League.
On 22 October 1969, Mulligan signed for English club Chelsea for £17,500. He featured in Chelsea's UEFA Cup Winners' Cup success in 1971, making a late substitute appearance in the first final against Real Madrid in Athens. He also played in Chelsea's League Cup final against Stoke City a year later, though they lost 2 - 1 at Wembley.
Mulligan left Chelsea in 1972 for Crystal Palace. He stayed with the South London club for three years, after which he signed for West Brom. Mulligan re-joined Shamrock Rovers in 1979, making his final appearance on 24 February 1980. Has his testimonial in August 1980 . He then joined Panathinaikos as assistant manager before retiring from the game in 1982.
During his time at Rovers he won 10 caps for the Irish national team, including three as captain, one under-23 cap and six Inter league caps. Overall he won 50 caps for his country.
Mulligan managed League of Ireland sides Galway United (during the 1983 - 84 season) and Shelbourne during the first half of the 1985 - 86 season. He also applied to manage the Irish national team, but lost out to Eoin Hand.
Mulligan occasionally appears as a soccer analyst on Irish television channel TV3.