@Andrew Firstly, apologies for the delay in responding here!
We do not have any such ready(built-in) feature for your scenario, you can make use of Events to trigger a function app whenever a new file is uploaded and based on that trigger they will have to have their own logic to transfer the file to an external server.
If you wish you may leave your feedback here, All the feedback you share in these forums will be monitored and reviewed by the Microsoft engineering teams responsible for building Azure.
Please let us know if you have any further queries. I’m happy to assist you further.
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