The Anonymous User access to sites has been throrolly discussed, but the contribution of an Anonynnous User to a list or Library should be an easy thing to do . . . just set the right options and there you go! or maybe not!
There is a small detail that can contribute to this failure . . . it happened to me. If you choose the template Publishing Site for your Site Collection, the Anonymous User won’t be able to contribute.
Explanation: well I just don’t know but my problem was just that.
I found out the solution because I happened to find an FAQ from Mike Walsh, that answered that this was a problem identified by Microsoft.
Solution: If you have to have Anonymous contribution choose a diferent template 🙂
You can find this FAQ here.