You can use this API REST (https://learn.microsoft.com/en-us/rest/api/recoveryservices/usages/list-by-vaults)
The API will return a response in JSON with the information that you require
Also, you can review this thread https://learn.microsoft.com/en-us/answers/questions/871037/get-the-azure-recovery-vault-size-via-the-azure-re.html
Hope this helps!
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