Is there a plugin that is like a suggestion box?

You can set that they don’t have to fill in an email address to comment and a name they can choose themselves, so how more anonymous do you wan to have it? The admin not able to see IPs?

I want it so they can submit a comment like you said, but it not show up to the public?

I needed something similar for my company’s internal blog, but what I did was to set up a normal email contact form (I used Contact Form 7, even when I usually use Cforms II, just because I think it was better for the job). Then I removed the name, email and any other field that asks for personally identifiable info. In a similar way, I modified the email template the plugin sends, removing the sender’s IP, name, email address, etc.

Now, our HR Dept receives anonymous emails with suggestions and comments from employees. But for this to work without leaving an open door for anyone to write (and obviously, to spam), I must tell you that our blog allows access to members only, thanks to a clever combination of several plugins, with Members Only, Blue Steel Theme, Role Scoper and Disable Revisions and Autosave, among them.
I don’t know if this info solves anything, but maybe it can give someone some ideas…