An admitted company is an insurance company that is authorized and licensed to do business in a given state. So a NJ admitted insurance company is one that is licensed and authorized to do business in New Jersey.
Insurance companies are regulated by state agencies. To find out if in New Jersey an insurance company is allowed to exclude drivers on a business policy, check with the New Jersey Department of Banking and Insurance.
If a household member or resident relative of an employee can drive a commercially insured vehicle for their own personal use is likely up to the rules set forth by the business that pays the vehicle's insurance and the terms of the business or commercial auto insurance policy. If you cannot tell from the terms of the car insurance policy if this is allowable or not, contact the insurance provider and request information on the subject.