A former employee left the firm to take another job. He left no resignation letter just said he had another job.
The company esop sent his his put option papers and He now says he retired from the company. He was in the plan for 6-years, was fully vested and had reached retirement age when he left. When does the ESOP have to begin making payments? The company would like to wait 5 years.

Based on the facts you laid out, he is retired. Payments need to begin no later than 1 year after the close of the plan year of retirement. Do they think they can make him wait longer just b/c they don't like the way he left?

The Plan Document will say when distribution must be made to a participant who has terminated employment and has attained the age that the Plan defines as retirement age.

In some Plans, the retired person is eligible for a distribution very soon, for example, as soon as administratively practicable after the first of the month following or coincident with her/his termination date. Your Plan may say something else, but whatever it says, that's the thing to do.