Hi Tommie van Lent - Thanks for reaching out over Q&A Forum.
Yes, in order to start with investigation as to what is accessing the account, it is suggested to enable the diagnostic logging. Thereafter, we have to narrow down based on the available fields such as Client IP, URI, UserAgent header etc.
From the snippet you shared, it appears the user agent is some .NET SDK and container URL point to ASR, so probably something related to Azure Site Recovery but you have to re-verify that once. Also for the Private IP concern, that could be because of either resource in same region access storage regions and hence the call via internal backbone or some VNET/PE.
Please let me know if there are any further queries/concerns, will be glad to assist.
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