So you are in a hybrid environment and moved your Public Folders to o365 but suddenly 3rd party people can no longer send you emails?

Yeah… try this:

# For all Public Folders
Get-PublicFolder -Recurse | 
Add-PublicFolderClientPermission -User Anonymous -AccessRights CreateItems

# For a specific Public Folder
Get-PublicFolder -identity "\Folder" | 
Add-PublicFolderClientPermission -User Anonymous -AccessRights CreateItems