Kvile moved FK Fyllingsdalen to SK Brann as a youth player, also moving on to Fana IL and briefly making his senior debut before joining the junior ranks of Sarpsborg 08. In the summer of 2020 he finally made his senior breakthrough as he joined Åsane, and impressed enough to be picked up by reigning league champions FK Bodø/Glimt in 2021. Kvile made his Eliteserien debut in May 2021 against Tromsø.[2]
Kvile started 2022 by playing two matches in the postponed 2021–22 Norwegian Football Cup, as well as three of the 2021–22 UEFA Europa Conference League.[3] On 6 May 2022 he was loaned to Eliteserien competitors Kristiansund for the remainder of May.[4] Already in his first game, he sustained an injury, and had to undergo surgery at Haukeland Hospital.[5] Upon returning from injury, he featured once for Bodø/Glimt.[3] On 30 September, he was loaned to Kristiansund a second time.[6]
On 31 August 2023, Kvile moved to OBOS-ligaen side Fredrikstad on loan for the rest of 2023, with option to make it permanent after the season.[7] When Fredrikstad won the 1. divisjon that season, the deal automatically became permanent from January 1.[1]