Born in Cartagena, Elitim moved to Cali at the age of 12, joining Club JC FC Cyclones' youth setup. After impressing in a tournament at the age of 15, he agreed to a deal with Granada CF, but was unable to join the club until the age of 18; he was assigned to Leones FC instead.[2]
Elitim made his first team debut for Leones on 9 October 2016, aged 17, by playing the last eight minutes of a 2–1 Categoría Primera B home win against Tigres.[3] In July 2017, he moved to Granada after his 18th birthday.
On 22 August 2018, Elitim joined Segunda División B side Marbella FC for the 2018–19 season.[4] On 31 July of the following year, after having his federative rights assigned to Watford, his loan was renewed for a further year.[5]
On 9 July 2021, Elitim returned to Spain, joining Deportivo de La Coruña on a season-long loan deal.[8] On 2 July of the following year, he moved to Racing de Santander on a one-year loan deal.[9]
^"El duelo de felinos fue para Leones" [The duel of felines end up being of Leones] (in Spanish). Código Fútbol. 9 October 2016. Retrieved 20 September 2020.