EEOICPA Ombudsman Extension and Enhancement Act of 2009 - Amends the Energy Employees Occupational Illness Compensation Program Act of 2000 (EEOICPA) to: (1) make permanent the authority of the Office of the Ombudsman of the Department of Labor; and (2) increase the duties of the Ombudsman under the Energy Employees Occupational Illness Compensation Program, including to assist in making claims for benefits for Department of Energy (DOE) contractor employees who have contracted a covered illness through exposure at a DOE facility and to act as an advocate for such employees in appropriate cases.