[2] He began skating at a rink owned by the brother of former Capitals scout Vojtech Kucera before moving away from home at 15 to play in the Swedish Hockey League (SHL).

Upon making his Elitserien debut in October 2012, Vrána became the youngest foreign player in the history of the league and the ninth-youngest overall.

[6] At the conclusion of the 2013–14 season, Vrána was ranked fourth overall among European skaters by the NHL Central Scouting Bureau.

[1] During the 2016 off-season, Vrána worked on gaining weight and returned to the Capitals development camp 10 pounds heavier.

[3] In his second season in North America, Vrána once again began with the Hershey Bears but received his first NHL call-up on 29 November 2016.

[10] He subsequently made his NHL debut on 1 December against the New York Islanders and skated on the Capitals second line with Evgeny Kuznetsov and André Burakovsky.

[22] He scored the opening goal in game five of the 2018 Stanley Cup Finals against the Vegas Golden Knights which resulted in the Capitals' victory.

"[28] In January 2023, the Red Wings placed Vrána on waivers for the purpose of assigning him to the team's AHL affiliate, the Grand Rapids Griffins.

[29] On 3 March 2023, the Red Wings traded Vrána to the St. Louis Blues in exchange for prospect Dylan McLaughlin and a seventh-round pick in the 2025 NHL Entry Draft.

[32] Vrána would continue being a healthy scratch afterwards, and would play two more games for the Blues before being loaned to their AHL affiliate, the Springfield Thunderbirds.

On 15 August 2024, Vrána agreed to an invitation to return to his original club, the Washington Capitals, on a professional tryout agreement (PTO) for their 2024–25 training camp.