The North Reading, Massachusetts native finished his collegiate career with 144 points (80 goals, 64 assists) in 128 games and was named ECAC Player of the Year after the 2014–15 season.

Vesey was good enough for Nashville general manager David Poile to guarantee the senior a roster spot for the end of the 2015–16 regular season and playoffs.

[6] On March 30, 2016, almost five months before he was officially set to become a free agent, it was reported that Vesey would sign with his hometown team, the Boston Bruins.

On June 20, 2016, the Buffalo Sabres acquired Vesey's rights from the Predators in exchange for a third-round pick in the 2016 NHL Entry Draft.

[1] The deal was made in order to give the Sabres an extra few weeks of exclusive negotiating rights with Vesey before he became a free agent.

[8] While discussions with the Sabres continued, a deal was not struck and in accordance with the NHL CBA, Vesey became an unrestricted free agent on August 16.

[19] On March 12, 2018, Vesey recorded his first career NHL hat trick in a game against the Carolina Hurricanes, which the Rangers won 6–3.

[22] In the pandemic-delayed 2020–21 season, Vesey initially appeared for the Maple Leafs on the second line with John Tavares and William Nylander before shifting into in a bottom-six forward role, contributing with 5 goals and 7 points through 30 games.

On March 17, 2021, Vesey was claimed off waivers from Toronto by the Vancouver Canucks, marking his fourth NHL club in three seasons.

As a free agent from the Canucks, on September 14, 2021, Vesey was signed to a professional tryout contract (PTO) by the New Jersey Devils.