Yes. You can configure the JSCAPE MFT server so that the files that are uploaded are automatically streamed to an Azure File Share/Cloud Storage Service. Some options for configuring your JSCAPE MFT Server include uploading files on a predefined schedule, automatically redirecting each file uploaded to your server (for example, via FTPS, SFTP, HTTPS, etc.) to the share, encrypting each file before uploading to the share, or  selecting incoming files based on source, type, or some other criteria and then upload only those files to the share.
Learn how to use Azure as the storage system on JSCAPE