Bödvarsson started his senior career in the Icelandic second tier with Selfoss, before joining Norwegian side Viking a few years later. He later signed with German side FC Kaiserslautern. In 2016, Bödvarsson made a move to England, and signed with Wolverhampton Wanderers. He joined Reading a year later, followed by a move to fellow Championship side Millwall.

Bödvarsson regularly features for the senior team of Iceland and has represented his country at the UEFA Euro 2016 and 2018 FIFA World Cup.

On 20 January 2022, Bödvarsson joined Bolton Wanderers and ended the season on 7 goals.

In the summer of 2023, the striker extended his stay with the Whites by a further year.