Provisionally signed by Partick Thistle as a teenager and "farmed out" to Ayrshire Junior side Dalry Thistle, Wright signed full professional forms with Partick on Friday 28 January 1949 and made his debut the following day, exactly seven weeks after his eighteenth birthday, in the Glasgow derby against Rangers at Ibrox Park. Wright performed for Partick Thistle most of his playing career, appearing in three Scottish League Cup Finals which all ended in defeat. Wright subsequently played season 1963–64 for East Fife before having to retire due to knee injuries. He became a coach, working briefly for Clyde and Partick Thistle, before being appointed manager of St. Mirren at the age of 35, in October 1966.
St. Mirren were relegated at the end of the 1966–67 season, but Wright guided them to promotion back to the top flight at the first attempt. After four years at Love Street, Wright became manager of Dunfermline Athletic in October 1970. He held that position for only 16 months, however, as the club struggled to win games during a period of severe financial hardship which almost led to the Club's demise.
Following a brief period as assistant manager of Dumbarton, Wright was appointed manager in late 1972, a post he held for four years. Despite the Club having spent the previous 50 seasons in Division 2 prior to his appointment, Wright guided them to high of tenth position in the Scottish League in 1973–74 and a Scottish Cup semi-final appearance in 1975–76. He was involved with Dumbarton in various roles for 18 years. After leaving Boghead Park in 1990, Wright then worked as Chief Scout for Kilmarnock under managers Alex Totten and Bobby Williamson as well Scottish Scout for Millwall and Bolton under Bruce Rioch, and Leeds under David O'Leary.