Lawson was born and raised in Mount Brydges, Ontario. He is the youngest of three children and has two sisters, Sara and Kyla. His father Mike coached him throughout his minor hockey career, and mother Kristen is a principal and was born in Thunder Bay, Ontario.
Crouse was selected by the Kingston Frontenacs of the OHL in the first round, fifth overall, in the 2013 OHL Priority Selection. In the 2013–14 season, Crouse appeared in his first career OHL game on September 19, 2013, scoring his first career goal in that game against Andrew D'Agostini in the Frontenacs 11–4 win over the Peterborough Petes. On October 4, 2013, Crouse recorded his first multi-goal game of his junior career, scoring twice in a 7–4 win over the Ottawa 67's. On March 4, 2014, Crouse had a season high four point game, scoring two goals and adding two assists, as Kingston lost to the Barrie Colts 7–5. Overall, Crouse finished the 2013–14 season with 15 goals and 27 points in 63 games, helping the Frontenacs into the playoffs. In seven post-season games, Crouse was limited to one assist, as the Frontenacs were upset by the Peterborough Petes in the first round.
Crouse improved during his second season with the Fronteancs in 2014–15. He recorded his first career OHL hat trick on November 28, 2014, scoring three goals against Dawson Carty of the Kitchener Rangers in a 5–0 Frontenacs victory. He played in 56 games with Kingston, scoring 29 goals and 51 points. In four playoff games, Crouse had two goals and three points.
Crouse returned to the Frontenacs for a third season in 2015-16, and Crouse once again saw an improvement in his offensive numbers. On March 5, Crouse recorded a hat trick, scoring three goals and adding an assist in a 7-2 win over the Flint Firebirds. Overall, in 49 games, Crouse scored 23 goals and 62 points. In the post-season, Crouse scored seven goals and 11 points in nine games.
Crouse was drafted eleventh overall in the 2015 NHL Entry Draft by the Florida Panthers. On July 15, 2015, the Panthers signed Crouse to a three-year entry level contract.
The Panthers assigned Crouse to his junior hockey team, the Kingston Frontenacs, for the 2015–16 season, however, when the Frontenacs were eliminated from the post-season, Florida assigned Crouse to their AHL affiliate, the Portland Pirates, for the remainder of the season. In his first career AHL game, Crouse was held off the scoresheet in a 3-2 loss to the Springfield Falcons. Overall, he appeared in two games with the Pirates, earning no points.
On August 25, 2016, Crouse was traded along with Dave Bolland to the Arizona Coyotes for a conditional 2017 second round pick and a 2018 third round pick. Following an impressive training camp and pre-season, Crouse made the Coyotes regular season roster. On October 15, Crouse made his NHL debut, as he was held off the score sheet in a 4-3 victory over the Philadelphia Flyers. On November 1, Crouse scored his first career NHL goal against Martin Jones of the San Jose Sharks in a 3-2 win.