SharePoint discussion boards can allow people to anonymously post new entries in any individual forum. A few settings need to be configured and these include web application in Central Admin, anonymous for discussion list and users can further be configured to add/write to only own posts.
1) First go to the discussion board in your SharePoint site:
2) Go to the List Settings under the List tab in the top ribbon.
3) Select Permissions for this discussion board under List Settings.
4) Select Stop Inheriting Permissions under Permissions tab.
5) Select Anonymous Access under Permissions tab.
6) Select what level of access you want to give as shown in the screenshot:
7) Now you can post as an anonymous user.
8) Your post will appear as any other posts as shown below: