Gaunce, a 6-foot-2 forward, was selected by the Vancouver Canucks in the first round (No. 26) of the 2012 NHL Draft after he had 68 points (28 goals, 40 assists) in his second season with Belleville of the Ontario Hockey League. On Aug. 13, 2012, he signed a three-year, entry-level contract with the Canucks.

The Sudbury, Ontario, native was Belleville's captain until he was traded to Erie on Nov. 26, 2013. In 43 games with Erie, he had 46 points (21 goals, 25 assists), finishing the season with 72 points (31 goals, 41 assists) in 65 games between the two teams.

Gaunce played his first professional season in 2014-15 with Utica of the American Hockey League, where he had 29 points (11 goals, 18 assists) in 74 games. He was recalled by Vancouver on Oct. 28, 2015, made his NHL debut the following night against the Dallas Stars and scored his first career goal the next night in a 4-3 win against the Arizona Coyotes before he was sent back down to Utica.

On March 1, 2016, he was recalled by the Canucks and finished the season with one goal in 20 games.

Gaunce spent most of the 2016-17 season with Vancouver, appearing in 57 games and finishing with five points, all assists, while leading all Canucks forwards with 94 hits. On Aug. 9, 2017, Vancouver re-signed him to a two-year, one-way contract.

Gaunce had six points (four goals, two assists) in 37 games during 2017-18 but missed time because of a foot injury.