Rømer started playing football for Køge Boldklub alongside his brother Marcel, before moving to the FC Midtjylland academy as a 15-year-old.[1][2] In March 2012, Rømer signed a five-year contract with Midtjylland, and he was permanently promoted to the first team.[3] He made his debut on 26 September 2012 in a 1–0 win over Rishøj Boldklub in the Danish Cup.[4] Rømer made his Danish Superliga debut on 9 December 2012 in a 2–0 win over AC Horsens, where he came on as a substitute in the 90th minute for Petter Andersson.[5] In total, Rømer made seven league appearances and played one cup match during the 2012–13 season.[6] At the end of the season, he was also named "Newcomer of the Year" in Midtjylland.[7]
Rømer scored his first goal on 18 April 2014 in a 4–0 win over AGF, after coming on as a substitute in the 85th minute.[8] He played a total of 22 league matches and scored one goal, with an additional two cup appearances during the 2013–14 season.[6] In October 2014, Rømer extended his contract with Midtjylland until the summer of 2019.[2] He played 28 competitive matches and scored a goal during the 2014–15 season when Midtjylland became league champions.[9][6] During the season, Rømer made his UEFA Europa League debut in the play-off round against Greek club Panathinaikos.[10][11]
In the 2015–16 season, Rømer played 41 competitive matches and scored one goal.[6] He played in the qualifying rounds for the UEFA Champions League and in the Europa League, where Midtjylland reached the round of 16 before they were knocked out by Manchester United.[6][12] In September 2016, Rømer extended his contract with Midtjylland until 2021.[13] In the 2016–17 season, he played 41 competitive matches and scored three goals.[6] During the autumn of the 2017–18 season, Rømer played 11 competitive matches.[6]
OB
On 4 January 2018, Rømer signed a three-and-a-half-year contract with league rivals OB.[1] Rømer made his debut on 16 February 2018 in a 1–2 loss against FC Nordsjælland.[14][15] He played 11 matches for the club during the 2017–18 season.[6]
Randers
On 19 July 2018, Rømer moved to Randers FC on a three-year deal.[16] Rømer played 35 competitive matches and scored five goals during the 2018–19 season.[6] The following season, he played 34 competitive games and scored two goals.[6] In the first six months of the 2020–21 season, before leaving the club, Rømer played 19 competitive matches and scored four goals.[6]