At a minimum, your business must carry Workers Compensation Insurance and Unemployment Insurance. Tennessee does not require Disability Insurance (although several states do require it), but there are guidelines for voluntarily carrying Disability Insurance.
Tennessee requires businesses with 5 or more employees to carry Workers Compensation Insurance. It does not matter whether the employees are full or part time or even seasonal. AND it does not matter if the employees are family members. There are some exceptions! Construction companies, coal mining enterprises are required to carry Workers Compensation Insurance with only 1 employee.
Tennessee requires a business with one employee who is employed for 20 weeks of any calendar year or who is paid $1500 or if you have acquired a business that was already paying UI taxes or you are required to pay Federal unemployment taxes and have one employee in Tennessee. To find out if you are liable for UI taxes, you should apply for an account with the Tennessee Department of Labor. https://www.tn.gov/workforce/employers/tax-and-insurance-redirect/unemployment-insurance-tax.html
Not required by Tennessee guidance for voluntary coverage can be found here: https://www.blr.com/HR-Employment/Benefits-Leave/Disability-Insurance-in-Tennessee