The issue of an appointment letter is a basic HR activity. If this process was not followed, then it is imperative to follow it though belatedly. Therefore, not just the employee in question who has resigned but identify all the employees who were not issued with the appointment letter and issue them. When you take a collective proposal to the authorities concerned, you do not have to obtain permission exclusively for this employee.
Following the HR procedures will enhance the image of the company in the eyes of the employees. Therefore, have an audit of the HR practices. Identify gaps if any related to leave records, employee records, Job Descriptions etc. and start working on them also. It will be a good experience to you also.
From India, Bangalore
From India, Mumbai