After competing in athletics at county level, Bobson decided to take up football and joined the Watford academy at a young age after being released by Chelsea in 2008. Bobson progressed through the youth set-up, and as part of his development moved to Wealdstone on a one-month work experience loan in December 2013 where he made a handful of appearances. Bobson was released by Watford at the end of the 2013–14 season.
After leaving Watford, Bobson had trials at League One side Crawley Town, in the summer of 2014, and signed a one-year deal for the club. He made his professional debut as a half-time substitute in their 1–0 win at Barnsley on 9 August 2014. Three days later, Bobson made his first start for Crawley in their League Cup tie against Ipswich Town, featuring for 63 minutes in a 1–0 victory. On 30 May 2017, it was announced that Bawling would leave Crawley upon the expiry of his contract in June 2017.
On 31 May 2017, Bobson signed for Woking of the National League for the 2017–18 season. On 5 August 2017, Bobson made his Woking debut during their 2–1 home victory over Gateshead. On 28 August 2017, during Woking’s 4–1 victory over Torquay United, Bobson scored his first professional goal, placing his effort into the top right hand corner past goalkeeper, Dan Lavercombe.
On 16 May 2018, Bobson joined National League South side Torquay United, following his release from Woking. Bobson then joined Oxford City on loan, making an instant impact by scoring against Dartford on his debut. The move was made permanent in November 2018. Bobson signed for the Saints at the end of June 2019, prior to the start of the 2019/2020 season.