Born in Ironbridge, Shropshire, he played in the wing-half and other defensive positions. Wright was the first ever player to represent his country a hundred times. He captained England during their campaigns at the 1950, 1954 and 1958 World Cup finals.
His association with Wolves began in 1938 when he was taken on as a member of their ground staff. He was only 14 years old when he made his debut for Wolves in a B team game against Walsall Wood in the Walsall Minor League after being accepted on an eight month trial by Major Frank Buckley. He made his first team debut for the club aged just 15 in a 2-1 win at Notts County in 1939. This game was played shortly after the start of WW2, so is not counted as the official debut. His official debut came in the 1945/46 FA Cup in a 2 legged tie against Lovells Athletic.
He signed as a professional at 17, but the deepening effects of World War II soon forced Wolves to suspend competitive football. Wright turned out as a guest for Leicester City, playing as both a forward and a defender before he returned to Molineux in 1942. He joined the army in 1943 as a Physical Training Instructor, playing for Wolves whenever possible, making over 100 appearances in wartime football.
He became club captain soon after the end of the conflict, with the playing retirement of Stan Cullis. With Wright leading the team, they won the First Division title three times (1954, 1958 and 1959) as well as the FA Cup in 1949. He was a virtual ever-present, missing only 31 games for Wolves during the 1950s.