The General Data Protection Regulation (GDPR) gives people the right to know on what grounds computer programs make decisions.

what为疑问限定词，修饰ground，what ground作介词on的宾语。on+疑问限定词+名词引导间接疑问句作know的宾语从句。介词on在句首为正式语体，介词on 留在句末为非正式语体。二种都是正确的，只是语体风格不同而已。类似的句子有：

Which room do you live in? 非正式语体，日常常用。

In which room do you live? 正式语体，只用于正式书面语。