I have previously unlocked all the badges from the badge pack that is available on some event (from this box:
https://sto.fandom.com/wiki/First_Contact_Day_Badge_Box ), i.e. Federation Klingon Badge, Andorian Badge, Vulcan Badge, ... However, the Federation Insignia is no longer available. I know that I have unlocked it because I am using it on one character but it is no longer available to me on any other character.
Comments
Most badges are limited by what uniform you pick, so if it's not showing up, it might be due to your choice of jacket.