The club was founded 18 December 1993 as FS Vágar. It was a merger of Vágar island teams MB Miðvágur and SÍF Sandavágur in order to increase the footballing standard of this part of the Faroe Islands. SÍ Sørvágur, also located on Vágar, joined the club in 1998. Soon, the newly founded club promoted to the national top league which, at that point in 1995, was called 1. deild. Unfortunately, the team was not able to permanently secure a top-tier spot over the years. In 2003, FS Vágar was relegated for good. Soon after that, the alliance between the three founding clubs began to crumble, and FS Vágar was eventually disbanded in the fall of 2004.
Despite the controversy, many people wanted to keep the club alive or, if this was not possible, found another club. So, on 8 November 2004, the club was re-founded as FS Vágar 2004 (FSV04). In fall 2007, talks about a merger between FSV04 and SÍ Sørvágur, one of the members of the first incarnation of the club, started and were successfully concluded on 6 November 2007. The club was renamed 07 Vestur, shortly afterwards. The new name refers both to the founding year of the new club and the location of Vágar island, which is approximately 7° W.
The club currently maintains two men's teams and also a women's team. In 2009 the men's first team played in the Faroe Islands Premier League, but they were relegated and played in 1. deild in 2010. They won 1. deild and were promoted to the Premier League; in 2011 they were playing in the Premier League of Faroese football, but again they remained there for only one season, they ended as 9th with 24 points and got relegated to 1. deild. The first-team recently gained promotion to Effodeildin in 2012 after winning 1. deild with 68 points.