Microcosm

• Paracausal Beam: a Kinetic beam with massive bonus damage to shields.
• Paracausal Imbuement: final blows grant Super energy, and the weapon gains bonus damage once your Super expires.
• Fixed parts: Hammer-Forged Rifling, Enhanced Battery and a Hand-Laid Stock.
• Catalyst: kills add a little Super energy, and the Kinetic ordnance staggers combatants more easily, tracks targets better and hits weakened targets harder.

• Rootbound: find the injured Ghost, investigate the new growth in the Arbor of Light, and speak with Micah.
• Underbrush: acquire a Flickering Blessing mod from chests in the Pale Heart and equip it.
• Greenery: open Overthrow chests in the Landing, fight the infection in the depths of the Pale Heart, and return to the Arbor.
• Budding: complete campaign missions on Cooperative Focus mode, then return to the Arbor.

• Difficulty: Specific Quest Steps, or Full Convalescence Quest; the steps above can be ticked one by one.
• Extras: Complete Destined Heroes Pt. 1, and Self-play (Sherpa).

• The Final Shape expansion with its Pale Heart destination. Destiny 2: The Collection includes it.

Does Bungie list the quest as available?

Yes. The new-player path for The Final Shape names Convalescence: Budding among the Exotic quests.

Where is the rifle sold?

The vendor data also lists it under Micah's Conduit.

Which ammo does it use?

Heavy ammo, as a Kinetic trace rifle.