Hi Ed,
This is not possible and frankly it doesn’t sound like a very good idea to me. Handling private keys should be done with care.
Can you detail an example of where this could be useful?
You need to help your users understand that if they deposit Bitcoin to your site, they will NOT be getting any bitcoin forked airdrops such as Bitcoin Cash, Gold, etc. Either they have their coins on their own wallet, or you are holding their coins or their behalf.
P.S. By the way, it’s just me here, see https://www.dashed-slug.net/dashed-slug/team/ No need to say hi to anyone else!)