Other jurisdictions not so much. There are various scrapers available to obtain generic race card information[2] and APIs from some bookmakers/exchanges.[3]
A couple of years ago I read an article lamenting the lack of open data in horse racing, which still largely holds true today. Unfortunately, I can't find the article now (it was either on the Paulick Report or Bloodhorse website), but here's a more recent one from May this year calling for more open data.[4]
In fact, a question was asked on HN a couple of years ago about an open horse racing database and as far as I know the answers there are still valid.[5]
In general, there is data available for US, UK, IRE, Australian racing (the main jurisdictions) but they have to be paid for.[6][7][8][9]
[1] https://racing.hkjc.com/racing/english/index.aspx
[2] https://github.com/4A47/rpscrape
[3] https://docs.developer.betfair.com/display/1smk3cen4v3lu3yom...
[4] https://www.bloodhorse.com/horse-racing/articles/232487/thor...
[5] https://news.ycombinator.com/item?id=7779117
[6] https://www.betwise.co.uk/smartform
[7] https://www.proformracing.com/