Where to find Dratini in Sun, Moon, Ultra Sun & Ultra Moon

SOS Encounter: Poni Gauntlet, Poni Meadow, Vast Poni Canyon (Only in Sun & Moon)
Super Rod: Poni Gauntlet (1%), Poni Meadow (1%), Vast Poni Canyon (1%, Only in Sun & Moon)
Surf: Poni Gauntlet (10%), Poni Meadow (10%), Vast Poni Canyon (10%, Only in Sun & Moon)

Dratini also appears in

How to get Dratini in Sun, Moon, Ultra Sun & Ultra Moon

Its base catch rate is 45 out of 255 (higher is easier); the catch rate calculator works out your odds with each ball. For the look, the best colour match is Dive Ball.

Evolution: Dratini → Dragonair (Level 30) → Dragonite (Level 55).

Dratini in every game · Pokémon Sun, Moon, Ultra Sun & Ultra Moon interactive map (Alola)