Depends on what the organization does. They cannot campaign for someone, but they can try to influence.
Under the Internal Revenue Code, tax-exempt organizations are allowed to try to influence the nomination or confirmation of a potential justice to the federal court, including the Supreme Court. However, the rules on such lobbying vary, depending on the section of the Code under which the exempt organization operates.
Section 527 political organizations. Unlimited lobbying to influence Senate confirmation of judicial appointments by section 527 organizations is permitted. Under the Code, exempt function activity for political organizations includes expenditures for the purpose of influencing the appointment of an individual to public office.
501©(3) charitable, etc., organizations. Limited lobbying to influence Senate confirmation of judicial appointments is permitted.