As a youth, he played in the 1978
Quebec International Pee-Wee Hockey Tournament with a
minor ice hockey team from Boston. He began attending
Merrimack College in 1984 and played there for four years. He had an instant impact with the
Merrimack Warriors, scoring 19 goals and 11 assists in his freshman year. During his sophomore year Vesey played 33 games and scored 29 goals and 33 assists, and was named an
All-American for the first time, and he was also the first sophomore in program history to receive the honor. During his junior year he helped Merrimack win their first
ECAC East championship, with a 58 point season with 22 goals and 36 assists, and Vesey was again named an All-American. During his senior year at Merrimack, Vesey was named team captain and played in all 40 games, helping lead the Warriors to another ECAC East title, as well as their first appearance in the NCAA
Division 1 tournament in 1988. They went on to pull off a huge upset over
Northeastern in the first round, before ultimately losing to eventual national champions
Lake Superior in the quarterfinals. At the conclusion of the season Vesey was once again named an All-American for a third consecutive time. He was also be named the 1988 Division II-III
Hobey Baker Award winner after having a 95 point season with 40 goals and 55 assists. This is still the program record for most points in a season as of 2025. Vesey left Merrimack as the program leader in both assists and point. Drafted 155th overall by the
St. Louis Blues in the
1984 NHL entry draft, Vesey went to play 11 games for the Blues, scoring a goal and two assists. He had three stints with in the Blues'
International Hockey League (IHL) team the
Peoria Rivermen. In the
1988–89 season, Vesey was one of the leagues top players, scoring 93 points (47 goals 46 assists), and was named to the IHL First All Star team, while also being named Peoria's team MVP. In
1989–90 Vesey scored 47 goals for the second year in a row, on top of 44 assists, and during the
1990–91 IHL season he recorded 73 points. Vesey played in 19 playoff games for Peoria that season, scoring 4 goals and 14 assists, and the Rivermen won the
Turner Cup. Vesey finished with 263 points for Peoria over three seasons. Vesey was then traded to the
Winnipeg Jets to complete a February 28, 1991, trade in which Winnipeg had sent
Tom Draper to St. Louis in exchange for future considerations. Vesey was sent to Winnipeg on May 24, 1991. Less than one month later, Winnipeg traded Vesey to the
Boston Bruins in exchange for future considerations on June 20, 1991. He signed with Boston in 1991 and played four games for them, scoring no points. Vesey played two seasons for the Bruins'
American Hockey League (AHL) affiliate, the
Providence Bruins, in his first season only appearing in 10 games. During the
1992–93 season Vesey played 71 games for Providence, achieving a 77 point season with 38 goals and 39 assists. He would end his two years in the AHL with 100 points. Vesey scored the first goal in Providence Bruins history at 4:06 of first period on October 6, 1992. The following year Vesey returned to the IHL, signing with the
Phoenix Roadrunners, scoring 30 goals and putting up 40 assists. Vesey retired after the 1994–95 season, ending his IHL career with 156 goals and 171 assists. ==Post playing career==