Vesey played four years with Harvard University in the NCAA. The North Reading, Massachusetts native finished his collegiate career with 144 points in 128 games and was named ECAC Player of the Year after the 2014–15 season. In his junior year, Vesey's outstanding play was rewarded with a top-ten nomination for the Hobey Baker Award. On April 2, 2015, Vesey was named to the Hobey Hat Trick, the 3 finalists for the award, along with University of North Dakota's Zane McIntyre and Boston University's Jack Eichel, with Eichel going on to win the award. In 2016, Vesey once again qualified for the Hobey Hat Trick, along with Michigan Wolverine'sKyle Connor and Boston College's Thatcher Demko, this time going on to win the Hobey Baker Award on April 8, 2016.
Professional
Vesey was selected by the Nashville Predators in the 3rd round with the 66th overall selection of the 2012 NHL Entry Draft. 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. But Vesey informed the Predators he was not interested in signing. Vesey wanted to choose his destination, which any collegian can do four years out from when they are drafted. Vesey's decision came as a huge blow to the Predators' organization. However, it was also within the bounds of the NHL's Collective Bargaining Agreement, which gives teams four years to sign college draftees and thus allows for the drafted player to move to free agency shortly after his senior year. 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. 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. Despite this, Vesey's agent informed ESPN that he had still intended to become a free agent. 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. In the two days following his free agency, Vesey met with seven NHL teams. Five more teams requested to meet with Vesey, but were denied.
On August 20, 2016, Vesey signed with the New York Rangers. After attending the training camp, Vesey received the Lars-Erik Sjöberg Award, which is given to the best Rangers' rookie in the camp. On October 17, 2016, Vesey scored his first NHL goal in a 7–4 win over the San Jose Sharks. On November 8, 2017, in a game against the Boston Bruins, Vesey scored 2 goals in 29 seconds. This was the quickest number of goals scored by a Rangers player since Jaromír Jágr did so in 26 seconds in 2006. The Rangers eventually won the game 4–2. 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.
Buffalo Sabres
On July 1, 2019, the Sabres again acquired Vesey in exchange for another third-round draft selection through a trade with the Rangers. This marks the first time in professional sports that a player was traded to the same team with the same compensation twice.
His father, Jim Vesey, was drafted by the St. Louis Blues in the eighth round of the 1984 NHL Entry Draft, going on to play 15 games in the National Hockey League; he works as a scout for the Maple Leafs. Jimmy's younger brother Nolan was drafted by the Toronto Maple Leafs in the 2014 NHL Entry Draft. He was traded in June 2018 to the Edmonton Oilers and signed an entry-level contract with them. Growing up, Vesey was friends with 2012 NHL Entry Draft pick Matt Grzelcyk. The two first met when they were around the age of six, playing hockey together for a team named the Middlesex Islanders. The Islanders were coached by Vesey's father. They similarly both attended the Belmont Hill School in Belmont, Massachusetts, and played hockey together there.
