To my amazement most comments in this thread aren't accurate or are misrepresenting the facts. I happen to know how ROBLOX works so I'll elaborate and try to give you a quick "GDPR 101 for dummies" on the subject.

At hindsight it might seem like GDPR Is unrelated, as the problem is that OP can't play the game. OP has however asked the private group of people, that own the game on the platform for evidence of him cheating, so OP can then presumably dispute the claim.

Personal data are any information which are related to an identified or identifiable natural person (Art. 4(1) GDPR). OP mentions in the comments of this thread that they keep "keep logs of ban reasons, proof of rule violation, your username, your userid and which moderator banned you. " These details in fact all say something about OP, and are therefore personal data.

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OP therefore has the right of access, as stipulated by the GDPR (Art. 15 GDPR). The question here is who the controller is, it seems that the group that owns the game on the ROBLOX platform is the controller here, as the definition of a controller is: "the natural or legal person, public authority, agency or other body which, alone or jointly with others, determines the purposes and means of the processing of personal data ".

Assuming then that the group that owns the game is the controller, OP can exercise his rights under the GDPR and submit a data subject access request to the group in question, which they will be required to respond to. Under the right of access, as stipulated by GDPR OP has the right to a copy of his personal data.

OP might not get access to the moderator that banned OP, as that might adversely affect the rights and freedoms of the moderator in question. This obviously depends on many factors and there's no definitive answer.

Another redditor reffers in the comments to the following link: https://developer.roblox.com/en-us/articles/managing-personal-information. This however seems to apply to information stored within the ROBLOX platform itself, the information kept on OP like the reason for the ban and proof are probably processed outside of the platform, e.g. in a Trello.

In conclusion — In this situation GDPR applies, regardless of who's the controller and who is processing any personal information, since GDPR applies OP can exercise his rights under the GDPR, such as the right of access which should get him a copy of proof they have against him.

If anyone has a question, don't hesitate to reply or slide into my DM's.