While the method Araghar mentioned used to work, Blizzard does not supply this luxury anymore. Any request regarding it will be met with some generic copy-paste reply about how they can’t breach accounts like that anymore.
It’s unfortunately a lot of chance and luck involved now. They only release names if the account (not just the character) hasn’t been logged in for two expansions.
Tazjin, if the name is still taken but not showing up on the armory, it is likely taken by a character lower than level ten. Have you tried to mail them perhaps? Additionally, you can try your luck here: Wyrmrest Name Archive Redux