Ögmundur made his senior debut for Fram on 12 June 2006, a week before his 17th birthday, in the 2006 season of the Icelandic second tier, starting against Haukar in the absence of Gunnar Sigurðsson. He kept a clean sheet in a 3–0 away win. His next match came in the 2009 Icelandic Premier League season in the last match of the season, entering as a half-time substitute for Hannes Þór Halldórsson and keeping a clean sheet in the second half as Fram lost 1–0 to Þróttur R. at home. He remained a reserve goalkeeper to Hannes Þór Halldórsson in the 2010 season. When Hannes transferred to KR after the 2010 season Ögmundur became the main goalkeeper for Fram. He has played every league game since the start of the 2011 season. On 30 December 2011, Ögmundur was given the honorary title "Fram Sportsman of the Year", the first footballer to hold that title.
After the 2013 season he went on a trial to Norwegian club Sandnes Ulf and Scottish club Motherwell. His trial at Motherwell went well, although they were not looking to buy a goalkeeper at that time. On 16 July 2014, half-way through the Icelandic 2014 season, Ögmundur joined Danish club Randers, signing a one-year contract. During the 2014–15 Superliga season, Ögmundur only made four league appearances while mostly acting as a back-up to Karl-Johan Johnsson.
Hammarby
On 2 June 2015 Ögmundur signed a 3-year deal with Hammarby IF, eventually making 15 appearances in the 2015 Allsvenskan for his new side. Ögmundur replaced Johannes Hopf as Hammarby's first choice goalkeeper halfway through the season, after the latter chose to make a move abroad. In 2016, he scored his first competitive goal in his senior career when he scored the deciding penalty in the Swedish Cup semifinal against local rivals AIK. Ögmundur played all 30 league games as Hammarby finished 11th in the table, keeping 7 clean sheets throughout the year. He started the 2017 season as Hammarby's starting goalkeeper, under the reign of new manager Jakob Michelsen. Ögmundur was criticized by the latter in May 2017 after several weak performances. In late July the same year, Hammarby brought in Johan Wiland from reigning champions Malmö FF. Ögmundur was deemed surplus to requirements after the new acquisition.
On 2 August 2018, Greek Superleague club AEL officially announced the signing of Kristinsson on a two-year deal. On 26 February 2020, he signed a two-year contract extension.
Olympiacos
On 7 July 2020, Kristinsson signed with Greek Superleague club Olympiacos for 400 thousand euros.
International career
Ögmundur made his first international appearance on 4 June 2014 in a match against Estonia, when he came on as a substitute for Gunnleifur Gunnleifsson at half-time. In that second half, he kept a clean sheet, to maintain Iceland's 1–0 lead. On 13 October 2015, Ögmundur made his competitive debut in a UEFA Euro 2016 qualifying game against Turkey, with Iceland already being qualified for the championship. Despite putting on a solid effort, Iceland went on to lose with the score of 1–0 after a free kick goal by Selçuk İnan in the 89th minute. He was selected for EURO 2016.
Personal life
Ögmundur has a bachelor of law and has stated that he wishes to become a lawyer after his football career. His favourite team is Manchester United whose former Danish goalkeeper Peter Schmeichel was his childhood hero.
