Like many of his generation, Stoneman began in karting before stepping into single-seaters via the Formula Renault BARC series in 2006. He competed full-time in Formula Renault 2.0 UK from 2007, finishing as runner-up to Hywel Lloyd that year. He continued in the series in 2008 and 2009, taking multiple wins but finishing fourth in the standings on both occasions. His performances earned him a nomination for the McLaren Autosport BRDC Award.
Stoneman won the FIA Formula Two Championship in 2010, producing a campaign strong enough to earn him a test drive with the Williams Formula One team at the Abu Dhabi young drivers test at the end of the season. He set the fifth fastest time on the first day, approximately two seconds off pace-setter Daniel Ricciardo.
In 2011, Stoneman had been scheduled to progress to the Formula Renault 3.5 Series with ISR Racing alongside Ricciardo. However, he was diagnosed with testicular cancer and was forced to withdraw from the entire season. During 2012, while regaining fitness, he competed in powerboat racing — a family tradition, as his father Colin had previously been a champion — and won the P1 SuperStock UK powerboat title.
Stoneman returned to single-seater competition in late 2013, driving for Koiranen GP in the final two rounds of the GP3 Series season. For 2014 he signed with Marussia Manor Racing for the full GP3 campaign and raced in the first seven race weekends before the team folded. He switched to Koiranen GP for the final four rounds, replacing Carmen Jordá, and finished the season as runner-up in the drivers' championship — a remarkable result given the disruption to his programme.
In 2015 Stoneman stepped up to the Formula Renault 3.5 Series with DAMS under Red Bull Junior Team backing, finishing sixth in the standings with 130 points. He joined GP2 with Carlin for the final three events of that season, picking up a single championship point and finishing 24th overall.
Stoneman moved to Indy Lights in 2016 with Andretti Autosport, finishing fifth in the championship and testing an Andretti Dallara IR12 IndyCar during the year. He returned to sports car racing in 2017 with Strakka Racing, sharing a McLaren 650 GT3 with Andrew Watson and Jazeman Jaafar in the Blancpain GT Series Endurance Cup.
Stoneman's career arc — a dominant Formula Two title, a near-fatal illness, and a determined return to competitive racing — marks him as one of the more resilient figures of his generation in British motorsport. His runner-up finish in GP3 in 2014 demonstrated that his ability had not diminished despite the years away from single-seater competition.