Is it not possible to rename an Azure Storage Table?
I cannot seem to find anything online (not even cmdlets). There are no options for this in Visual Studio Server Explorer, Cloud Storage Studio or TableXplorer.
You're correct. It is not possible to rename an Azure Storage Table (or Blob Container or Queue for that matter).
Possible solution would be to download all entities from the table and upload them again in another table. Once all entities are uploaded, you can then delete the old table. When downloading entities, please do keep Continuation Token in mind as querying table would return up to 1000 entities per request.
You can download all entities using either Cloud Storage Studio (or Azure Management Studio) from Cerebrata or TableXplorer. If you want, you can use Azure Management Cmdlets from Cerebrata as well. It has cmdlets to export a table (Export-Table) and restore a table (Restore-Table).