Should the chair of the board of directors be allowed to initiate investigations into weaknesses in ...

The investigation of internal control weaknesses is usually a management function; however, as stated in the COSO integrated framework Management is accountable to the board of directors, which provides governance, guidance and oversight. A strong, active board, particularly when coupled with effective upward communications channels and capable financial, legal and internal audit functions, is often best able to identify and correct such a problem..
The provisions of SOX Section 302 require CEOs and CFOs to certify that they are responsible for internal controls and have evaluated the company's internal controls. Nonetheless, in this case it seems that the chair had the responsibility to investigate the leak of strategic information given that some of the suspects were members of the board of directors.
Under the circumstances, it would have been prudent for Ms. Dunn to share her plan with the Executive Committee of the board and get their guidance and blessing.