Chychrun started playing hockey in his home state of Florida with the Junior Everblades and the Florida Jr. Panthers organizations. When Chychrun was 13, he joined the Little Caesars hockey organization in Detroit, Michigan, where he played very well during his major bantam season. At the time, he continued living and practising in Florida, travelling to Michigan on the weekends to play. The following season he remained with Little Caesars, playing for their U16 team. In 2013, Chychrun was drafted by the Youngstown Phantoms in the USHL draft. USA Hockey implemented a rule requiring 15-year-old players to receive a special exemption to play junior hockey. Chychrun applied for one but was denied. As a result, Chychrun moved to Toronto and played his minor midget season for the Toronto Jr. Canadiens in the Greater Toronto Hockey League.
Major junior
Chychrun was selected first overall by the Sarnia Sting in the 2014 OHL Priority Selection. He had a strong rookie season for the Sting and at the end of the season was nominated for the OHL rookie of the year award, i.e. the Emms Family Award; OHL defenceman of the year, i.e. the Max Kaminsky Trophy; and OHL Player of the Year, i.e., the Red Tilson Trophy. Leading up to the 2016 NHL Entry Draft, Chychrun was highly regarded by scouts and was considered one of the top defencemen available. He was drafted by the Arizona Coyotes in the first round, sixteenth overall. This was far lower than expected; prior to the start of the 2015–16 season, Chychrun was regarded as a top three selection.
Professional
On July 30, 2016, Chychrun signed a three-year, NHL entry-level contract with the Coyotes. He scored his first NHL goal on October 20, 2016, against Carey Price of the Montreal Canadiens. After a month with the Coyotes, the Coyotes announced Chychrun would remain with the team for the rest of the season. He finished the season with seven goals and 20 points in 68 games. On August 3, 2017, Chychrun underwent knee surgery after suffering an injury during offseason training. On November 28, he was assigned to the Coyotes' American Hockey League affiliate, the Tucson Roadrunners for conditioning purposes. Chychrun made his NHL season debut on December 3, scoring a goal in a 3–2 loss to the Vegas Golden Knights. He was ruled out for the remainder of the season on April 4 after suffering a lower-body injury. He finished the season with four goals and 14 points in 50 games. Chychrun underwent surgery to repair a torn ACL in the 2018 offseason. Prior to his season debut on November 13, the Coyotes signed Chychrun to a six-year, $27.6 million contract extension. He finished the season with five goals and 20 points in 53 games.
International play
Chychrun competed with Team Ontario at the 2014 World U-17 Hockey Challenge where he scored four points in five games. He was the first minor midget player to play for Team Ontario at the tournament.
