Onsham was the first man from Thailand to play in the Masters, playing in the 1970 and 1971 events. He missed the cut both years. He is often referred to as the first non-white person to play at the Masters; this is false. The Japanese golfers Torakichi Nakamura and Koichi Ono played in the 1958 Masters Tournament well before Onsham. Also several other players from Japan, Taiwan, and the Philippines played in the event during the 1960s.
Onsham had much success in 1972 on the international circuit. Early in the year at the Malaysian Open, he finished one behind Japan's Takashi Murakami, tying Walter Godfrey and American Marty Bohen for second. On 23 October 1972, Onsham finished runner-up to American Bob Murphy at the 1972 Wills Masters, an event on the Australian Tour held outside of Sydney. Two weeks later he posted another runner-up in Australia at the Dunlop International, four shots behind Tony Jacklin, finishing tied for second with David Graham. That year he also placed third at the Japan Open Golf Championship.
Onsham's career began to wind down in the 1980s, however, he did have some success that decade. He finished runner-up at the 1981 Indonesia Open. A young Payne Stewart made a 20 foot birdie putt to defeat him and Taiwanese golfers Hsu Chi-San and Chen Tze-chung on the first playoff hole. As of 1988, he was a golf instructor.
Onsham also represented Thailand multiple times in the World Cup.
- "Sukree Onsham". Asian Senior Masters Golf. Retrieved 16 June 2019.
- Buranaphan, Siriporn (29 May 1988). "Links Are a Lure to Clients : In Bangkok, Golf Is Very Big Business". Los Angeles Times. Reuters. Retrieved 16 June 2019.
- "Taiwan, US lead golf". The Canberra Times. 3 October 1969. p. 22. Retrieved 9 January 2020.
- April 5, 2006. "Fit to be Thai Jaidee with Masters Incentive". Golf Channel. Associated Press. Retrieved 16 June 2019.
- "Did Clifford Roberts Say "As Long as I'm Alive, Golfers Will be White, and Caddies Will be Black"?". Sports Urban Legends Revealed!. 23 August 2012. Retrieved 17 June 2019.
- "Men Asia tournament results". where2golf.com.
- "Bob Murphy wins Wills Masters". The Canberra Times. 23 October 1972. p. 14. Retrieved 17 June 2019.
- "Jacklin takes Dunlop title". The Canberra Times. 6 November 1972. p. 14. Retrieved 16 June 2019.
- "Malaysian Open won by Thai". The Canberra Times. 1 April 1974. p. 14. Retrieved 14 January 2020.
- "Stewart wins Indonesian Open in play-off". The Canberra Times. 7 April 1981. p. 21. Retrieved 8 January 2020.
- "Rahmat has the last laugh". Singapore Monitor. 19 December 1983. p. 32. Retrieved 12 February 2020 – via National Library Board (Singapore).