After attending King Edward VII School in Sheffield,[2] van Oostrum started his basketball career with British basketball league club Sheffield Sharks in 2008. In May 2009, van Oostrum moved to Spanish ACB team Caja Laboral,[3] who loaned him to the reserve team Euskotren Fundación Baskonia, of the fifth tier of the Spanish basketball league.[4]
In the summer of 2012, Baskonia sent him on loan to the LEB Oro team Cáceres Patrimonio de la Humanidad.[6] In February 2013, he came back to the club of Vitoria after the departure of Carlos Cabezas.[7] In September 2014, he was loaned to Cibona Zagreb for one season.[8] On 31 October 2014, Cibona terminated contract with him, so he returned to Laboral Kutxa.[9]
On 14 November 2014, Baskonia agreed a new loan, this time to Finnish team Kouvot.[10] On 22 December 2014, he parted ways with Kouvot after playing 6 games with the club.[11] On 7 January 2015 he was loaned to Leeds Force of the British Basketball League[12] and made his debut three days later, scoring 16 points and 6 assists shooting 6-17 from the field on a loss against Plymouth Raiders.[13] Afterwards he was loaned to Peñas Huesca of the LEB Oro.[14][15]
On 11 August 2016, Van Oostrum signed with Koroivos Amaliadas for the 2016–17 season.[18] On 13 January 2017, he was released by Koroivos.[19] Seven days later, he signed with Macedonian club MZT Skopje for the rest of the season[20] On 27 February 2017, he parted ways with MZT Skopje because of knee injury during a semifinal match in the Macedonian Cup.[21]
In August 2019, van Oostrum signed a two-month contract with CBet Prienai of the Lithuanian Basketball League (LKL).[25] Club had an option to extend the contract for remaining of the season but decided against it and player was released on 27 October 2019.[26]
On 4 December 2019, Van Oostrum returned to the Netherlands to sign with Landstede Hammers of the Dutch Basketball League (DBL).[27] On the Hammers, he joined his older brother Nigel who has been on the team since 2016.
On 23 November 2020, Van Oostrum was sent on loan to Peñas Huesca for four weeks because the DBL had been suspended due to the COVID-19 pandemic.[28] In February 2021, it was announced he had to miss the remainder of the 2020–21 season after a surgery.[29]