It could be done BUT it would require changing a few database tables as well as modifying some php files.
First create the "canpostanonymously" permission. Do this by adding a field of that name to mybb_forumpermissions. Change admin/forumpermissions.php to include a field for this permission. Finally, add "canpostanonymously" to $fpermfields inside inc/init.php .
Next, add a field "anonymous" to mybb_posts. Check that field every time you need to add a new post (including new threads) or display a post. Create an anonymous postbit to use in all situations where the "anonymous" flag is set and the poster has proper permissions.
Finally, use the ACP to assign this permission to the proper groups based on forums.
Additionally, it would be prudent to change the "find all posts by this user" to NOT return anonymous posts as that would definitely give away his / her identity.
Sounds simple. LOL
I'm not a coder. Oh well, maybe someday someone will come up with this modification.
