No, you do not need to complete a Form I-9 for an employee who was hired before November 7, 1986. The Immigration Reform and Control Act (IRCA), which established the requirement for employers to verify the identity and employment authorization of their employees using Form I-9, did not go into effect until November 6, 1986. Employees who were continuously employed by the same employer since before November 7, 1986, are exempt from the Form I-9 requirements.
As long as the employee in question has been continuously employed with your company since July 1986, you do not need to complete a Form I-9 for them. However, for any employees hired on or after November 7, 1986, you must ensure that a Form I-9 is completed and kept on file.