Update
Azure Blob Storage now supports MSI (Managed Service Identity) for "keyless" authentication scenarios! See the list of supported services here.
Old Answer
Unfortunately Blob Storage is not supported, either to have it's own identity or to provide access to services that have their own identity. The reason is because Blob Storage (all of Azure Storage) does not work with Azure Active Directory.
However you do have other options, like Azure Data Lake Store or SQL Server that can be accessed via MSI.
Also note that Key Vault is supported. In your case, I would create a SAS key to the storage with the relevant restrictions in place and then place the SAS key in Key Vault. Use MSI from your WebApp to retrieve the key from Key Vault. Interestingly, we have been able to build this configuration out using ARM templates, placing the trust in the WebApp MSI and the storage key in the Key Vault, with only the deployment account ever knowing the key, nice and secure, developers never see or need the SAS key...
You can find the list of services that support MSI (Managed Service Identity) at this link.