Thanks Szymon, 

Yes, I have the exact set up as the above and the access connectors are added to the allowed resource instances list and they have contributor role on the storage account. 

With the above set up, are you able to read/write to the storage account ? For me it is not working unless I "Enable from all network".

Not sure why I get this error message "Please check your Azure Firewall - Full error message: Your request failed with status FAILED: [BAD_REQUEST] This Azure storage request is not authorized. The storage account's 'Firewalls and virtual networks' settings may be blocking access to storage services. Please verify your Azure storage credentials or firewall exception settings"