Wregget played for three seasons with the Lethbridge Broncos of the Western Hockey League and won the Del Wilson Trophy as the WHL's top goaltender in 1984. In 1983, he joined the St. Catharines Saints, the Toronto AHL affiliate, after being drafted by the Toronto Maple Leafs in the third round of the 1982 NHL Entry Draft, 45th overall. He split time for two seasons between St. Catharines and the Maple Leafs. The 1986–87 season was his first full year in the NHL.