In order to enter the @ char, you'll need to press all keys required at the same time.

That is; instead of pressing first AltGr then pressing 2 while AltGr is down, you have to press both keys at the same time (as far as humanly possible).

Of course, the mail interface will disappear as a result of pressing the Alt key, but as soon as you interact with the Postal Courrier again, the @ char will sit safe and sound within the address text box.

Writing mail like this might be bothersome, but it's definitely less bothersome than having to copy-paste @ from another window or changing one's keyboard layout.
All that's left to do, is hoping for Cryptic to notice that not all players have American layout keyboards and fix that annoying design bug accordingly.

Edit:

I just noticed, this may be a bug in Neverwinter. According to the default keybindings, Toggle cursor mode is bound to Left Alt, and not to Alt in general. Doesn't make it less annoying...

Alternatively, binding the Toggle cursor mode to a different key may do the trick. Just tested it with Ctrl and it works perfectly with the AltGr + 2 combo. Thanks @Viper_Sb

Yes, it is possible to use the bank without being in a guild. When you approach the bank, the NPC in the center is the Guild Banker, used to access guild-only storage; to his left and right (in yellow below) are two other NPCs you can approach, who are the regular bankers, used to access your personal storage.