You don't need to prove that the account has been open for 15 years in order to file the return and process the distributions. You would only have to prove it if you are audited.
If you are audited then the way to prove it - since the original financial institution is no longer a viable institution - is through your personal records. Your bank will have records showing your contributions leaving your account back 15 years ago. You may have to pay the bank a fee to access those records (almost certainly) but the records exist. A contribution being sent from your account would prove that the 529 was in existence at that point. Also, statements were sent to the IRS for the 529 at that point 15 years ago. Accessing those statements is not a two foot putt but can be done.
@acask
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