Unnur started her senior team career with Haukar in the Úrvalsdeild kvenna in 2005 and won the national championship with the club in 2006 and 2007. After spending the 2008–2009 season with Salama Vaasa in Finland, she signed with Úrvalsdeild club KR in June 2018 She had an outstanding performances during the playoffs, helping KR advance to the Úrvalsdeild finals where they met Hamar in a best-of-five series. In game three of the finals series, she scored 33 points, making 13 of her 19 shots. In the fifth and deciding game of the series, she led KR to the championship, scoring a game high 27 points. For her performance, she was named the Úrvalsdeild Playoffs MVP. After the season she stepped away from basketball and moved to Hungary to pursue a medical degree. She returned to KR in 2017, helping the club win the second-tier 1. deild kvenna with a perfect 30-0 season and achieve promotion back to the Úrvalsdeild. On 6 February 2019, with KR in first place, Unnur Tara suffered an anterior cruciate ligament injury in a victory against Breiðablik and was initially ruled out for the rest of the season. She returned to the court on 20 March, scoring 14 points in KR's unexpected loss to last-place Breiðablik. In 22 appearances during the regular season, she averaged 9.1 points and 6.2 rebounds. In the playoffs, KR lost to eventual champions Valur 3-1. In the four games, Unnur averaged 5.8 points and 4.0 rebounds. In KR's second game of the 2019–20 season on 16 October, Unnur, who is a practicing medical doctor outside of basketball, was involved in a controversy with referee Ísak Ernir Kristinsson after she requested permission to enter the court during a stoppage to attend to a teammate who had been suffered a serious leg injury after a hard landing moments before. Ísak denied her the permission and gave her a technical foul after she indicated that she would then ask one of the other referees at the game for permission instead. After the incident caused an uproar with fans, the Icelandic Basketball Association Referee Committee issued a statement where it stated that Ísak admitted that it had been a mistake on his behalf to deny Unnur permission to attend to her injured teammate and that the Committee agreed with that assessment.
