The Tranquilizer is a single-purpose dart gun Sarah gives you to sedate a Newt. It is in Junk condition and only works for that task. You cannot keep it or use it in normal combat.
Sarah hands it over for one job: putting a Newt to sleep. Aim at the Newt and fire when the game prompts you.
It is not a weapon you can pick for normal fights. For quiet kills, use the Crossbow, Drifter Crossbow or Rock Chuck.
Crossbow
The Crossbow is an Average-condition special weapon for silent kills. It reloads slowly, but bolts make no noise. It gets more useful as you unlock more bolt recipes.
Drifter Crossbow
The Drifter Crossbow is the Military Spec upgrade to the standard crossbow. It is still silent, and it fires every bolt type: standard, Residue, Incendiary, Poison and Explosive. It is the best special weapon for stealth.
IPCA Stun Gun
The IPCA Stun Gun is a taser on the Survival Wheel. You earn it by collecting all 18 pieces of IPCA Tech. It stuns Freakers and sometimes sets them on fire, but both prongs have to hit.
Proximity Bomb
The Proximity Bomb is a crafted trap that goes off when enemies get close. Its recipe sits in the Berley Lake Ambush Camp bunker. Place them in a Horde's path or at a door you expect enemies to use.