I would kick this to the estate attorney since there are probably legal hoops to jump through.  My guess: the S Corp shares need to be transferred by probate, then the "new owners" (maybe wife or maybe wife and other heirs) have a shareholder meeting to elect a board or appoint an officer who can then issue a corporate resolution to change the signer on the bank account.  Also, likely a different path if this is an LLC electing to be taxed as an S Corp instead of an actual S Corp.  But these are all lawyer things, not accountant things.

For BOI, I remember seeing an announcement that changes have to be reported within 30 days of the settlement of the deceased's estate but I don't remember those rules contemplating the situation before the initial report is filed (another reason to get a legal opinion).