Here we have possible fault on three fronts. The first is the person who placed the item in the bank and put it in the wrong tab, thus making available to anyone a recipe that was supposed to be exclusive to a subset of the membership. The second is the person who took the item. The third is the person who set up the guidelines for the bank.
All of these issues could have been avoided with better communication. The person who deposited the item could have checked with an officer about what tab to place it in. The person who took the item could have checked with an officer before withdrawing and learning the recipe. And the officer who set up the bank guidelines may not have been clear about where things should go or what is acceptable to take without asking.
As officers, it's important to recognize that these communication lapses do not solely rest with the members in every case. More often than not, we are also at fault to a greater or lesser extent. So to the guild blaming, kicking, and demanding restitution from this person, I will say this: Take a look at what you could have done to avoid this situation in the first place.
You are treating this former member like a ninja. Maybe he is. I'm only getting his side of the story (with very little specifics) so it's impossible to tell if he was simply eager to withdraw something he could use or looking to make a profit from the guild's resources. But the officers involved may have drawn some dire conclusions about what could have been a simple misunderstanding.
To the author of the e-mail, your reputation has been tarnished on the server whether you are guilty or not. If you want to restore your good name, you'll have to settle things with your former guild. I suggest first attempting to speak with the guild leader about the situation. Don't get off on the wrong foot by pointing fingers at the other people involved. That's only going to cause more strife.
Instead, just express your regrets about the situation and tell your side of the story. Be calm and rational at all times. Having dealt with this myself many times, I am much more likely to side with someone who can present their case without getting overly emotional. Those who resort to anger or tears to convince me that they were wronged are often, in my experience, being less than truthful about what really happened.
I can fully understand being upset during or immediately after a conflict. After a day or two, however, I'm less interested in your feelings than I am in getting to the bottom of everything. "QQ" doesn't get much sympathy anywhere in Warcraft. Just spend 5 minutes on the official forums if you don't believe me!
If the guild leader doesn't accept your side of the story, like it or not you'll have to offer something in return for the recipe to restore your reputation. It may be very difficult for you to replace it if it comes from a raid, so an alternative may be necessary. Either way, if you don't ultimately make this incident right in their eyes, they could very well prevent you from finding another home by sharing their low opinion of you with other guilds' officers.
Next time, if you have any doubt at all, ask before you take. This axiom applies to loot in every circumstance. And officers, make your loot rules and policies as clear as possible. Better communication is the key here.