Prior to beginning his professional career, Vesey played 128 games over four seasons with Harvard University, collecting 144 points (80-64-144) and 66 penalty minutes. He was twice nominated for the Hobey Baker Award (2015, 2016), winning the award in 2016 against fellow nominees Thatcher Demko and Kyle Connor. Vesey was also named the ECAC Player of the Year in back-to-back seasons for 2014.15 and 2015.16.
A native of Boston, Massachusetts, Vesey was originally selected by Nashville Predators in the third round, 66th overall, at the 2012 NHL Entry Draft.
Vesey will wear jersey #24 when he joins the team.