IANAL and not particularly familiar with the particulars here, but very likely, the answer is "there is no legal recourse."

As a private entity, chess.com is within its rights to admit or reject people for any reason it wants, except on the basis of certain protected classes (which cheating is not one of them). Furthermore, the terms of use for an account probably says something to the effect of "we have the right to ban you for whatever reason we feel like, and you have no real recourse." One could still attempt to sue, but the almost certain result is to flush tens of thousands of dollars in the toilet just to get thrown out on the motion to dismiss for lack of a case.

They may have a slightly stronger case for defamation or similar because chess.com said the account was closed for "fair play violation", but still.