This information should be a part of your payroll system as a part of each employee's record. The department or group in your organization that handles the quarterly unemployment tax payments should also have access to this information.
As a Thomas & Company client you can be set up on our Tax Gateway platform so you have quick access to all of your unemployment account numbers in order to share them with your employees who need them in the claim filing process. Reach out to your Tax Analyst or Client Relationship Manager for more information about Tax Gateway or to get a current listing of your state and federal account numbers.