Search indexing documentation suggests in the Re-indexing a single project section that Project admins are able to reindex a project:

If you have Jira Project admin permissions, you can re-index a single project as follows:

1. In the bottom-right corner of the screen, select Project settings.
2. Under the Project settings (the left-side panel), select Re-index project.
3. Select Start project re-index.

This information is not true and only users with Jira admin permission (or Jira system admin) can see this option under the project settings.

Here is even a suggestion to include this functionality that was closed as "Won't Do": JRASERVER-40139.