Azure Blob Storage on IoT Edge module requires an internet connection to start up. If there is no internet connection, the module will not start. However, once the module is started, it can continue to run offline as long as it has access to the local storage.
If you need to use Azure Blob Storage on IoT Edge module in an offline environment, you can consider using Azure Blob Storage SDKs for .NET, Java, Python, and other languages to interact with Blob Storage directly from your code. This way, you can store and retrieve data from Blob Storage without requiring an internet connection.
You can find more information on how to use Azure Blob Storage SDKs.
Do let me know if you have any further queries.