So, we solved it today along with another developer.

The problem was very odd to say the least and it was coming from another package and not Umbraco Forms themselves.

Just for future reference for anyone searching: The error in the screenshot comes because Umbraco Forms cannot find the actual forms in either the App_Plugins OR (in my case) in the Azure Blob Storage.

So the real problem was the fact that the project was configured as I said to use Azure Blob Storage using a FileSystemProviders.config where it lists the providers:

 <Provider alias="forms" 
    type="Our.Umbraco.FileSystemProviders.Azure.AzureBlobFileSystem, 
    Our.Umbraco.FileSystemProviders.Azure">
<Parameters>
  <add key="containerName" value="form-data"/>
  <add key="rootUrl" value="https://myProject.blob.core.windows.net/form-data"/>
  <add key="connectionString" value="DefaultEndpointsProtocol=myConnectionString"/>
  <add key="maxDays" value="365"/>
  <add key="useDefaultRoute" value="true"/>
</Parameters>

So, this line was the problem: <add key="rootUrl" value="https://myProject.blob.core.windows.net/form-data"/>

Right above this line it mentions 'containerName' and it's set as 'form-data'. That's the name of the blob container the Umbraco Forms are in. Now, in the root url you'll notice that this is repeated in the url: https://myProject.blob.core.windows.net/form-data . This was the problem.

And I couldn't tell at first because whoever is using the same package that sets up forms and media to use with Azure, then you'll notice that if you do /media on the equivalent entry, it works fine.

Unfortunately doing /form-data (or whatever your container is that contains the forms) breaks the functionality. Remove it and just use the root url of the storage account https://myProject.blob.core.windows.net and it will fix the problem.