You can allow posting for anonymous users and automatically put those posts under moderation with the follwing settings:
- in Administration center - Permissions set "Asking questions" and "Answering questions" to "Anybody"
- in Administration center - Spam enable "Use moderation for anonymous posts"
However, these settings will enable anonymous posts for the entire site, not just one category. There is a plugin permissions2categories, but that seems to be mostly for limiting (view) access to selected categories to authenticated users with a given permission level, not elevating (post) access for unauthenticated users. I have never used it, though, so you'll have to try it yourself.