Elgezabal made his professional debut on 4 September 2016, starting in a 1–1 away draw against CD Numancia.[8] After contributing with 21 appearances as his side avoided relegation, he moved to the latter club on 16 August 2017, on a one-year loan deal.[9]
On 15 August 2018, Elgezabal renewed his contract until 2020 and returned to Alkor also in a one-year loan deal.[10] The following 6 July, after returning from loan, he terminated his contract with the Armeros,[11] and signed a permanent two-year deal with Alcorcón five days later.[12]
On 10 September 2020, Elgezabal terminated his contract with Alcorcón,[13] and signed for third division side Burgos CF fourteen days later.[14] On 18 June 2024, after four seasons as a starter for the latter which included a promotion to the second division in his first year, he joined Levante UD on a one-year deal.[15]