I found a way using DeviceCodeCredential. This will allow you to manually create a token with the identity of the user logged in to the Databricks workspace and using the notebook.
from azure.identity import DeviceCodeCredential
from azure.storage.blob import BlobServiceClient
tentant_id = <your tentant id>
credential = DeviceCodeCredential(tenant_id=tenant_id)
blob_service_client = BlobServiceClient(account_url=<your blob storage>, credential=credential)
This will print out a webpage where you will need to paste a code printed out to verify your identity. You can then use this credential to access a blog storage that is otherwise blocked for the workspace. You will not be able to automate using this method but it can allow you to get access to resources outside the permissions of the Databricks workspace.