I have a couple of Azure Automation runbooks that update Dynamic Distribution Lists. To make this work, I gave the Azure Automation System-Assigned Managed Identity permissions to access and make changes to Exchange Online using steps here; https://learn.microsoft.com/en-us/powershell/exchange/connect-exo-powershell-managed-identity?view=exchange-ps#step-4-grant-the-exchangemanageasapp-api-permission-for-the-managed-identity-to-call-exchange-online. These runbooks work as designed.