Richard started karting in 2011, driving in championships around Europe. In 3 years, he won a multitude of championships, ranging from 2nd in the Rotax Max Challenge Central-Eastern Europe Junior category, 1st in CIK-FIA Karting Academy Trophy, 2nd in Rotax Max Wintercup, Rotax Euro Challenge, WSK Champions Cup and WSK Final Cup, 3rd in CIK-FIA European KF Championship and won the German Kart Championship.
For 2015, he made a step up into single seaters, competing in a variety of Formula 4 championships. For the SMP F4 championship, he competed in all rounds, earning 339 points overall, stacked with 11 wins and 16 podiums. With these results, Richard won the championship ahead of fellow Dutchman Jarno Opmeer. Alongside SMP Formula 4, he competed in Spanish Formula 4, again dominating the championship with 17 wins and 19 podiums out of the 20 races, in which again Verschoor won the championship. At the same time, he competed in a few rounds of ADAC Formula 4 and Italian Formula 4. Here, he finished 15th and 21st respectively.
In the following year, he moved to Formula Renault, competing in Eurocup and the Northern European Cup. Both ended with him finishing in 9th, with him amassing a podium in Europcup and a win and 3 podiums in NEC. Also during 2017, he competed in Toyota Racing Series driving for Giles Motorsport. With 3 wins and 7 podiums out of the 15 races he competed in, Verschoor gained 843 points meaning he got 3rd.
In 2018, he competed once again in Formula Renault, this time competing mainly in Eurocup for 6 rounds and a few rounds of NEC, driving with Josef Kaufmann Racing. Out of the 12 races in Eurocup, he finished on the podium once, gaining 34 points in 13th. In NEC, he was a guest driver therefore he was ineligible to score points and place in the championship. Mid-way through the Eurocup season, Verschoor was called to MP Motorsport’s GP3 seat after Dorian Boccolacci went to replace Roberto Merhi in Formula 2. He finished 15th in the standings with a podium in 8 races, ahead of Finnish Simo Laaksonen. Once again, he competed in the Toyota Racing Series, this time with M2 Competition. Out of the 15 races, Richard won 6 and finished on the podium 12 times, earning 911 points. Despite this, he finished 2nd in the standings behind Russian Robert Shwartzman and ahead of New Zealander Marcus Armstrong.
FIA Formula 3 Championship
Verschoor made his Formula 3 debut in the inaugural championship of 2019, driving for MP Motorsport with Simo Laaksonen and Liam Lawson. His best position in 2019 was 4th in multiple races like Circuit Paul Ricard and Monza. In the end, he finished in 13th with 34 points.
Also during 2019, he competed in the Macau Grand Prix, an annual Formula 3 event. He was selected to drive with MP Motorsport with Liam Lawson and Lukas Dunner. Verschoor finished 4th in the qualifying race and ended up winning the main race, being the first Dutch driver to do so and on his debut.
2020 provides the second year of the Formula 3 championship, where he races again with MP Motorsport, this time with Bent Viscaal and Lukas Dunner. He will be driving as number 17.