After completing the research into a vaccine against the Scorched, the Vault 76 dwellers need to find the Fire Breathers. If they're gone, well, they need to pick up the torch and lead the fight against the scorchbeasts in the name of the Responders.
Joining the most presitigious units of the Responders requires completing several tasks. First is the knowledge exam, located at the Charleston Fire Department. The answers are:
Q1 - 3) Evacuate as quickly as possible
Q2 - 2) A water-soaked rag
Q3 - 1) Gently bind the burn with clean bandages
Q4 - 2) Retreat immediately
Q5 - 3) 1 pt. Purified Water, 2 Ash Rose, 2 Blight, 2 Soot Flower (as much as I wish it was #2)
Q6 - 2) Fall back and engage from a distance with firearms
Q7 - 3) End his life
Then the player needs to pass the physical exam (see Runner's High). After that's complete, they need to return to the knowledge terminal and initiate the final test: Find the Belching Betty, collect the Fire Breather's kit from the handler robot, and reach the bottom of the mine. The robot actually gives out a voucher that has to be redeemed at the nearby machine to obtain a free set of Fire Breather gear : Fire Breather uniform and Fire Breather helmet.
Reaching the bottom of the mine is easier said than done. The button is at the northwestern corner, on the platform with the dead Maxine Ballard and Rita Wilcox. Pressing the button will trigger an alert. Getting out is recommended, as it will attract any remaining scorched. Return to the Fire Breathers handler outside to receive the Welcome to the Fire Breathers holotape, then head back to the Charleston Fire Department and access the Master Terminal on the upper floor.
Finally, head on over to the dispatch room overlooking the fire engine hall and hit the button to complete the quest and hear Hank Madigan's message about finding a new way to fight off the scorched.