Born in Koper, Slovenia, Štulac started his career as a professional at hometown club Koper. He made his debut 2 April 2011 aged only 16 years and 6 months, making his debut in a 3–0 victory against Maribor. In 2013 he joined Dekani on loan where he scored 4 goals in 13 games, before returning to Koper. He won the Slovenian Cup during the 2014–15 season and the Slovenian Supercup in 2015. In 2016, he went on to play under Joe Balsamo at the world renowned Viareggio Cup. It was there, Leo was scouted and received a trial with Venezia FC.

 

On 15 July 2016, he joined the Italian Lega Pro side Venezia. In his first season he gained promotion to the Serie B. He also won the Coppa Italia Lega Pro with the club. Štulac scored his first Serie B goal on 24 October 2017, in a 2–1 away defeat against Cittadella. On May 5, 2018 he scored the winner in a 2–1 victory against Foggia. On 30 June 2018, Štulac signed for Serie A club Parma on a five-year contract, until 30 June 2023. 

 

Štulac has represented Slovenia at international level at all ages including Slovenia U21s. In March 2018, he received his first call-up to the senior national team for the two friendly matches against Austria and Belarus. He made his senior debut against Cyprus on 9 September 2018.