Making use of Azure Storage is your best bet. If your app is inside a docker container, you have the ability to mount Azure Files to windows hosted app or mount Azure storage to a linux hosted app. If you're not hosting your app inside a container, that's okay too. The decision you have to make is either using File or Blob.
With regards to security, Azure Storage offers encryption at rest and your application can easily access by using connection strings and storage keys that you retrieve from the portal.
For migration, I'm sure there are several ways to go about it. One example is simply downloading the artifacts from your kudu site; however, the first that comes to my mind would be to investigate utilizing a webjob as part of your app service that will retreive said artificats from your app service and upload them into your storage account.
Hope this helps.