Public folders are set to NOT allow external senders to public folders in Office 365. This can be fixed using the following receipt:
Conncect to the Windows Azure Active Directory Module for Windows PowerShell using the following commands.
Create a placeholder for your credentials:
$LiveCred = Get-Credential
Popup box will ask for your Office 365 Global Administrator credentials.
Create a placeholder for your Powershell Session towards Exchange Online.
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://ps.outlook.com/powershell/ -Credential $LiveCred -Authentication Basic -AllowRedirection
Import your Powershell Session to connect to Exchange Online.
Run this command to allow all scripts:
Then the public folder part
Add-PublicFolderClientPermission \(Path of the Public folder name) -User Anonymous -AccessRights CreateItems