Raised in Hamilton, Wright started his career with Aberdeen – progressing through the ranks alongside Eoin Jess and Scott Booth – and made 181 first team appearances for the "Dons"; he was part of the squad that came close to winning the League title on the last day of the Scottish Premier League in the 1990–91 season, and achieved further league runners-up finishes in 1992–93 and 1993–94, as well as reaching the 1992 Scottish League Cup Final and 1993 Scottish Cup Final, losing out to Rangers on every occasion. In 1994–95 the club's results were unexpectedly poor and they escaped relegation only via a play-off; these turned out to be Wright's last matches for the club, and the situation also meant his wedding took place in the short time between the regular season and the matches against Dunfermline Athletic rather than in a more relaxed atmosphere after its conclusion, as he had planned.