Are we obligated to pay for an hourly person that did not use the timecard system?

An employer is required to pay an employee for all hours it intstructed him to work and any hours it permitted him to work, even if the employee was not instructed to do so.  The answer to your question lies in whether or not the employee actually worked the hours he is claiming, regardless of whether or not they were accurately recorded on the time clock.  If he did the work, he is due the pay. Though the point is now moot because the employee has resigned, an employee who fails to adhere to the employer's instructions on accurately recording his time on the timeclock could be fired or otherwised disciplined, but he cannot be denied the pay (including any overtime pay) for the hours actually worked.