No, but VMs need access to Site Recovery URLs and IP ranges.

In reality, you are transferring the data across the Internet into Azure Site Recovery. But in many organisations, they don’t even allow Internet access on the servers, and they block it for security reasons. Because of that, there are certain URLs from Microsoft, so as long as we have access to those URL and they are whitelisted, then yes we can run the Azure Site Recovery and the service doesn’t need a separate Internet.