Kári came up from the junior ranks of Haukar and played his first games with the senior team in the 2013–2014 season, averaging 7.6 points in 25 regular season and playoffs games. The next season he moved to the starting lineup and averaged 15.0 points in 21 regular-season games. He averaged 13.0 points in 9 playoffs games, helping Haukar to the semi-finals where they lost to Tindastóll. He had his breakout season in 2015–2016 and was named the best player of the second half of the season before the playoffs. He helped Haukar advance to the Úrvalsdeild finals but was injured in the first game and missed the rest of the series which Haukar eventually lost 1–3. After the season he was named the Úrvalsdeild Young Player of the Year and to the Úrvalsdeild Domestic All-First team.
College
Kári joined Drexel University in 2016. On 26 December, he was named CAA Rookie of the week after scoring 17 points, including 5 three-pointers, and handing out 8 assists in a game against Quinnipiac University. For the season he averaged 10.1 points and 2.0 assists, starting 21 of 28 games. He led the CAA by making 43.6% of his three-point shots. He left Drexel in October 2017 prior to the start of the 2017–2018 college season, due to personal reasons.
Return to Iceland
On October 13, 2017, Kári signed with his former team, Haukar, for the rest of the 2017–18 season. On February 20, Kári broke his right thumb on a practice with the Icelandic national team, and was expected to miss up to 4 weeks, including the national team games and the final three games of the regular season. On March 8, Haukar defeated Valur in the last game of the regular season and finished with the best record in the league, winning the Division championship and a home court advantage through the playoffs. On March 20, 2018, Kári scored 6 of his 27 points in the last 3.4 seconds in game two of Haukar's first round series against Keflavík, giving them an 82–85 victory. After being fouled in the act of shooting with 3.4 seconds remaining, he made all three free throws and tied the game at 82–82. After a timeout by Keflavík, Haukar stole the inbound pass, allowing Kári to heave up a cross-court shot from his own free throw line that went in, winning the game for Haukar. On May 4, 2018, Kári was named to the Úrvalsdeild Karla Domestic All-First Team.
On August 3, 2018, Kári signed a one-year deal with Barça Lassa B of the LEB Oro. On 15 November it was reported that he would miss three months due to an Achilles tendon injury. Surgery was performed on both his ankles to remove a portion of his heelbones, to alleviate the pain it was causing to his Achilles tendons. Due to the injuries he was forced to miss the rest of the season.
In August 2019, Kári signed with Helsinki Seagulls of the Finnish Korisliiga. On 6 September, the Seagulls terminated their contract with Kári, before the start of the season, as he was not yet fully recovered from his injuries.
Haukar 2019–present
On 10 September 2019, Kári signed back with his hometown team of Haukar.
Kári is the son of Jón Arnar Ingvarsson, a former professional player and coach, who played 102 games for the Icelandic national basketball team. His nephew, Pétur Ingvarsson, played 26 games for the national team while his grandfather, Ingvar Jónsson, both played for and coached Haukar.
