Originally joining the club's academy at Under-15s level, David Robson signed his first professional contract with Hull City in December 2019 following the conclusion of his two-year scholarship.

In October 2018, Robson earned his first Wales cap with the Under-17s, playing the full game in a 2-2 draw against Belarus in the UEFA European Championship qualifiers.

Penning a new deal in January 2021, the goalkeeper has spent time on loan at Tadcaster Albion, East Yorkshire Carnegie and Gainsborough Trinity.

Robson was also an unused substitute in the Tigers’ 2020/21 EFL Trophy ties against Leicester City Under-21s and Fleetwood Town and the Championship fixture against Barnsley the following season.

Prior to the conclusion of the 2021/22 season, Robson helped Farsley Celtic stay up in the National League North, making eight appearances.

The shot-stopper made his senior Hull City debut in the EFL Cup fixture against Bradford City in August 2022 before moving to fellow League Two side Crawley Town on a six-month loan deal later that month.

Saving a penalty in a 8-7 shootout win over Portsmouth in the EFL Trophy on his Crawley debut, Robson made six appearances in total for the Red Devils, featuring three times in League Two and three times in the EFL Trophy.

Also making his Wales Under-21s debut in the 2022/23 season, Robson was rewarded with a new one-year contract in June 2023, with the club holding the option of an extra year. 

In the 2023/24 campaign, the goalkeeper spent a month on loan at National League North side Darlington, keeping one clean sheet in five games.