After a successful tournament at the Under 19 world cup in Malaysia, he started his domestic career for his home state of Punjab. A host of injuries derailed his smooth run and he was more out than in at the first-class level until 2012. However, then came the revival and Kaul has managed to spearhead the Punjab bowling line-up since then. Kaul made his debut for the Punjab cricket team against Orissa in the 2007-08 Ranji Trophy, alongside his brother who was wicket-keeper for the match. He took a five wicket haul in the first innings, and finished with figures of 5/97. He has also made appearances for the Punjab youth teams, at Under-15, Under-17 and Under-19 levels. He was the leading wicket-taker for Punjab in the 2018–19 Vijay Hazare Trophy, with twelve dismissals in five matches. In October 2018, he was named in India A's squad for the 2018–19 Deodhar Trophy. In October 2019, he was named in India A's squad for the 2019–20 Deodhar Trophy.
For the Indian Premier League's inaugural season, a number of members of the victorious India Under-19s team from the 2008 Under-19 Cricket World Cup and other specified youngsters were named as players to be drafted by teams in the Indian Premier League, whilst other youth players would have to be chosen to play for their local teams. Kaul was chosen by the Kolkata Knight Riders, based in Kolkata, West Bengal, and captained by Sourav Ganguly. Sunrisers Hyderabad bid for him in the 2016 season, but he was benched for that season. 2017 saw the skiddy pacer shooting into limelight for his 16 wickets in 10 games playing for SRH. He bowled some crucial overs for the David Warner led team and was an ideal foil to Bhuvneshwar Kumar. In January 2018, he was bought by the Sunrisers Hyderabad in the 2018 IPL auction. He also had a successful 2018 season being a crucial cog in the bowling Attack of SRH that took them to their 2nd finals. He was named in the IPL XI of the tournament by Cricbuzz.
Under-19 career
Kaul was named in the squad for the India U-19s team for the 2008 Under-19 World Cup in Malaysia. He played in all of India's matches as they proceeded to the final, where they defeated the South African U-19 cricket team by 12 runs. Siddarth was entrusted with the task of bowling the final over by his then skipper Virat Kohli against South Africa in Malaysia, where India won the under-19 World Cup for the second time. The skiddy pacer ended with 10 wickets from 5 games and jointly led the Indian bowling charts alongside Ravindra Jadeja, who was the vice-captain of that side. His 10 wickets came at an average of 15.40, placing him as the joint tenth on the list of highest wicket-takers for the tournament.
