Pretty Simple
Call something like anvil.util.take_screenshot()
This returns a screenshot image of the current app, that the user can save, download, etc…
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You can already do that with a single line of client side python code:
import anvil.js
screenshot = anvil.js.window.navigator.mediaDevices.getDisplayMedia()
Even if anvil did build this it would just be a wrapper around that.
As for screen recording, it bet that with a few minutes of googling you have some js lib ready to go. 
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