How do you write a letter to vendors and customers telling them that an employee no longer works for you?

You should try to send positive messages to your customers, so a better letter would be to announce the appointment of a new representative for the appropriate product, area or department. If you feel compelled to specifically state that the previous employee is gone, you could include a statement near the end of the letter that says: While we are sorry to see John Smith leave our organization after years of dedicated service, we are excited by the enthusiasm that Tom Jones brings to his new position allowing us to continue to provide the high level of service that you have come to expect from XYZ Company.