The Application for Authorization of an Emergency Vehicle in Tennessee is filed with the Legal Office of the Department of Safety, but it is not needed or routinely used by fire departments, rescue squads or public EMS agencies. Other parts of the Tennessee Code apply to those vehicles as long as the chief official of the fire department or rescue squad sign off for the individual responder and certify that to the local law enforcement official.
It used to be that if you responded outside of your home jurisdiction, then you needed the State authorization for liability protection to say you were an authorized emergency vehicle. For Fire, rescue, and EMS, the Mutual Aid and Assistance Act of 2004 and later amendments apparently eased that burden, because you are extended the same rights (when dispatched) that you have in your home locality. Trying to read and figure out the laws is a lot like filing an income tax form with the IRS. Hopefully you qualify locally and get to use the "short form" by getting the local letter. If certain other conditions apply, then you need to complete, "the long form". And it is always a good idea to get a rider on your own personal automobile liability insurance.
State agencies other than Dept. of Safety, TBI, Forestry, and TEMA, which are specifically addressed in the Code have had to file the State Dept. of Safety form, such as the State EMS office, fire marshal, TDOT, TOSHA, and commercial responders like haz mat clean up teams and industrial fire departments that have to use public roads. These organizations are insured differently (State Board of Claims, or self-insurance pools with high deductibles) and those factors have to be reviewed.
There may be a difference among individuals in registering depending upon the affiliations of departments. If you are only associated with one jurisdiction, then you may only need the local letter. If you serve with several agencies in different cities and counties, then you may need a State AEV authorization letter. When in doubt, contact the Legal Office of the Department of Safety.