Firstly, if you don't have the necessary Azure rights to view storage account information then no amount of APIs will work here - you don't have permissions. There is no workaround as that is how security works. So no role assignment = no access. You need the appropriate permissions and will need to work with your Azure infrastructure team to get it.
You can access storage using Entra ID, shared access signature or a shared key. You need to work with your Azure infrastructure team to get the necessary access based upon the approach they want you to take. Permissions in Azure are layered and additive so which permissions you need exactly can vary a little bit.
To access everything you could use Reader and Data Access
but that is a very high level and powerful permission. As such you'll generally be granted permissions to just the resources you need. For purposes of discussion we'll assume Entra ID authentication. If you use something else then permissions will vary.
To enumerate the storage accounts you must specify the subscription. The List Storage Accounts endpoint gives you that information. I cannot tell what permissions you need for this but I suspect it might be one of the storage account permissions defined here. Since this requires having access to a subscription it is going to need probably more permissions than the other calls.
To enumerate the containers in an account you use the List Containers endpoint. It is a simple GET request against the storage account. To call this API you need Storage Blob Data Reader
to the storage account itself.
To enumerate the blobs in a container you use the List Blobs endpoint. It also requires Storage Blob Data Reader
permission.
I'm not really sure what you mean by getting the blob's SAS URL. The SAS is a shared security so you must already know that. The List Blobs
endpoint gives you the information about how to get the blob via the API already.