Since the Export process relies on access to the file-system, this feature is not applicable in Artifactory Online. That being said, there are other options to transfer content from Artifactory Online to your local Artifactory instance.
The first option is to use the Retrieve Folder or Repository Archive REST query, which will allow download of an entire repository or a given folder within a repository.
It is also possible to create a script that performs a File List query for each of the repositories and then downloads the artifacts.
Both of the above will produce a folder structure on which you will be able to perform an Import procedure.
The second option is to set a replication process between the online instance and the local one. In order to do that, you need to create a corresponding repository on your local Artifactory instance for each repository you would like to copy/move from the Artifactory Online. You can either perform a Push Replication or a Pull replication.
For Push replication from the “local” online repositories, you will need to open your firewall to the Artifactory Online instance. For Pull replication, the repositories you create on your in-house Artifactory instance will have to be Remote Repositories. After bringing the content from the online Artifactory instance to a remote repository, you may copy/move the content to a local repository.