It is a Known Thing that you cannot extend permissions to anonymous users for Lists and Libraries in SharePoint online. Basically they won’t be able to contribute in Item Level (List/Library). Was the same in 2010 and yet remains in 2013 as well.
Just like the below figure in my scenario, Anonymous Users are only allowed to View Items from the Top Level and There’s no way to extend the level of Permission for the particular Item.
Breaking Inheritance and trying on Unique Permissions won’t even work here. So Extending Permissions to Anonymous Users in Item Level not feasible for the time being.
The trusted and easiest way to get this done is an WSP available in CodePlex which is a SandBox Solution – http://anonymous365.codeplex.com/ . I used this simply on my SharePoint Online and it really does what I needed.
To get this done, Simply download the above solution and upload it in to solution Gallery of your Site collection. (http://YouSiteName/_catalogs/solutions/Forms/AllItems.aspx)
Activate the Solution.
Create a List or library to Test the solution
Open up the Newly created Library/List and direct to Library/List on the Ribbon.
Pretty Simple as you can see below. Click on Anonymous access button created by the Solution.
That’s’ it and here it is. There will be several options that you can select up on the needs