Hello, I'm able to retag a question and therefore earn the organizer badge even if I have less than 500 reputation.
The retag privilege itself gives you the ability to retag questions without needing the approval of others. It adds a "retag" link next to the edit link that will allow you to edit only the tags, thus bypassing the whole suggested edit process. Clicking the edit button instead will take you to the normal editing screen where retags can still be made, but approval will still be required when assisted with other edits.
Note that while you have the retag privilege without full editing privileges, you cannot make a tag-only edit from the main edit screen. You must use the separate retag functionality.