I have a large blob container in one Azure Storage Account in one Directory (abc.com) that I would like to move to another Storage Account in a completely different Directory (efg.com).
Is the only way to do this via something like AzCopy? If so, should that be done in Azure somehow? (I assume that if I use AzCopy on my local machine I am downloading then uploading all the individual blobs?)
It seems one cannot move a resource between tenants/directories. However, you can transfer a subscription, so:
I assume what you are trying to do is to move one storage account from one subscription to another one that is in another tenant (Azure AD directory). If that's your case, then it is possible to move the entire storage account, assuming that you have no dependencies such as Virtual Machines in use with their disks within that storage account.
Just follow this guide below:
https://docs.microsoft.com/en-us/azure/azure-resource-manager/resource-group-move-resources#use-portal
Bear in mind that your user account in Azure must have admin rights to both subscriptions in order to be able to perform the move.
AzCopy is a server-side execution unless specified. The following command will allow you to copy all blobs within a Storage Account Container to another Storage Account (Specifying option /S uploads the contents of the specified container recursively.)
Link: https://docs.microsoft.com/en-us/azure/storage/common/storage-use-azcopy#copy-blobs-in-blob-storage