This is interesting question and we also came through with this issue.
We consulted MS for support for the same and in their reply we found that we can't share or give kind of permission to folder in App Service environment. Here below is response given to us my MS guy.
" the option to share a folder on the kudu console of your App Service with other App Services, Users and Virtual Machines isn’t available as the App service is a closed environment. I however advised, you can make use of a storage account to store files and have various App Services and VM’s access the storage. "
So, then after we created new Storage server and used it as a file sharing option in our different azure applications including our web app.
I will also try to write article here that how we shared folder in our application. But, hope this will help you to understand.