Typically, there are 3 aspects to the work with the Query Store, which can be reflected in roles:
1) Configuration – turning Query Store on and off, clearing the contents, flushing its contents to disk and changing its settings.
2) Viewing the reports or using the DMVs to analyze the Query Store contents (queries, plans and wait statistics) to gain insights but not necessarily having the authority to change anything
3) Actively change Plans by forcing or un-forcing, based on the information obtained from (2)
This is a nice overview of the problem and a fair amount of the solution.