@Dylan James , Thanks for reaching out. I have followed this documentation and added the resource instance as well. I can connect to storage account from logic apps using the managed identity.
Note: If your logic app and storage account are in the same region it is suggested to use HTTP trigger action as documented here. If they are in different regions, you can leverage the storage Blob trigger connector. Please verify if this is not the case.
Please follow up the documented steps and do let me know if you are still facing any issues.