Born in Stewart Street, Silksworth, Sunderland, his father Joe was a miner at Silksworth Colliery. His mother, Elizabeth, stayed at home to look after Bobby, his three brothers and one sister. Bobby took up football as a child, playing for his village team. His older brother, Ralph, also played football, as a goalkeeper. All his brothers went into pit jobs after leaving school.
Bobby was signed to Sunderland in May 1925, after being spotted by Charlie Buchan while playing for top non-league side Bishop Auckland. He made his debut nearly a year later against West Ham United on 3 April 1926, scoring once in a 3–2 defeat. He would play for the next three seasons alongside a striker who hit at least 35 league goals in each of his four full seasons at Roker Park, Dave Halliday, the most prolific goals to games striker in Sunderland's history.
After Halliday's departure Gurney was regularly the club's top goalscorer, garnering his best tally of 33 goals in the 1930–31 season. Among his career highlights were ten hat-tricks and two four-goal hauls. He was also one of just a handful of Sunderland players to score five times in a match.