The bullet requires 4 components, which are spread out on floors 2-5. Give these to the blacksmith on floor five.
The Prime Primer is purchased from the second floor shop. It costs 110 Shells and the pilot's shop discount passive does NOT apply to it. Try visiting the oubliette for extra shells if you are having trouble acquiring the money. You can also steal it using various items.
The Arcane Gunpowder is found on the third floor in a large room with floating minecarts. This room will spawn every run until you turn in the gunpowder. To cross, hop into the only accessible minecart and ride it along the path until you can dodgeroll into the next minecart. Continue on until you cross the chasm. Note that the Gunpowder is an active item (that has no effect) and you will have to abandon whatever active item you have to pick it up.
The Planar Lead is in a similar room on floor 4. It will spawn on every run until you turn in the Planar Lead. This room is a large pit with a random invisible path you must walk across in order to access the Planar Lead. Using your starting pistol, fire away from the pit. The spent casings will fall behind you and float on the invisible path which will illuminate the way. Any weapon that sprays liquid on the floor will also illuminate the way (Mega Douser, Fossilized gun) in a much easier manner. You can also hookshot or fly across if you happen to have those items.
The Obsidian Casing is obtained from the boss of the fifth floor. Once he is defeated, his skull will be laying on the left side of the room. Break this open with any gun and the Casing will drop from it.
You do not need to collect all four items in one run. Once you hand in any item, progress is saved and you never have to fetch it for her again.