Robinson began his career playing for KentSecond XI. After one match of the 2013 season, he left Kent for Leicestershire before playing for Yorkshire Second XI. He finished the 2013 season with 59 wickets and 1,282 runs in Second XI cricket, and made his first-team List A debut for Yorkshire in July 2013 against Leicestershire. In October 2013, Robinson signed a professional contract with Yorkshire. After making seven T20 Blast appearances for Yorkshire during the 2014 season Robinson was sacked by Yorkshire in July for "unprofessional actions". Yorkshire coach Jason Gillespie said of the situation that "When a player consistently displays behaviour that isn’t professional, there has to be a point in time when you say 'look, this isn’t really working, you’re obviously not that bothered about playing for the club'. That point came, and it was tough." and that "He’s got to learn about his game and about being a professional sportsman. I think there will be a future for him in the professional game, but I think he needs to learn a lot, and I think he can do it." Later in the 2014 season Robinson made one List A appearance for Hampshire. In April 2015, Sussex signed Robinson on a short-term deal, after an injury crisis with bowlers Tymal Mills, James Anyon and Lewis Hatchett all unavailable. Robinson had played a Second XI match for Sussex and was named in the squad for a County Championship match the next day against Durham where he made his first-class debut. In the match Robinson, batting at number nine, scored a century in a Sussex record-breaking tenth-wicket partnership with Matt Hobden of 164. In doing so, he became the first Sussex player in 95 years to score a century on their County Championship debut. Robinson took his first five-wicket haul in May 2015 against Warwickshire. In the 2015 season, Robinson took 46 Championship wickets at an average of 24.71, and was nominated for the LV= Breakthrough Player Award and won the Sussex Young Player of the Year award. In October 2015, Robinson signed a new three-year contract with Sussex. On 29 May 2020, Robinson was named in a 55-man group of players to begin training ahead of international fixtures starting in England following the COVID-19 pandemic. On 17 June 2020, Robinson was included in England's 30-man squad to start training behind closed doors for the Test series against the West Indies. On 4 July 2020, Robinson was named as one of the nine reserve players for the first Test match of the series. Robinson was then named in England's squad for the second Test match of the series, his maiden call-up to the senior team.
