Is there a way to get a random player that has played your game?

Sure, but the options here are also very limited.

If you were to save everyone’s user ID to a key, you would still have to sift through each key that contains user IDs in the case of a GDPR request. You can also only write to a single key once every few seconds so unless you’re batch applying updates to a key, you won’t be able to get the result you’re looking for very efficiently. And as OP said, they are only using one-person servers so making a system that allows a server to bulk apply multiple users’ user IDs from multiple servers to a single key is going to take a lot of work.

It’s also very possible you will run into other data store limits very quickly. 4 million characters per key isn’t a lot of space if your game blows up so you will very quickly run into needing to split up your data across multiple keys.

There is also no way to get a list of badges’ owners, with or without a proxy so that really isn’t an option either.
https://badges.roblox.com/docs#!/v1

I see. In that case, it would probably be better to use an ordered data store.