The bylaws of the agency would take precedence over Robert's Rules of Order Newly Revised . State and federal laws prevail over the bylaws. The by-laws state that normally, a vote carries with a majority of those present provided that there is a quorum. Bylaws often contain such provisions. The by-laws do not mention this.
Simple answer is no.
The board could pass a motion to have a mail ballot, which would enable those absent to cast a vote on an important question. This method is not a form of proxy voting, but a ballot procedure often used. The chair of the board is under the impression that the chairperson votes only in the case of a tie. The by-laws do not address this. Your web page suggests that this is not true.