Up-to-date as of the Indigo Disk:
Aipom with Double Hit (Ambipom)
Bonsly with Mimic (Sudowoodo)
Clobbopus with Taunt (Grapploct)
Dipplin with Dragon Cheer (Hydrapple)
Dunsparce with Hyper Drill (Dudunsparce)
Eevee with a Fairy type move, along with two hearts of Affection (Sylveon)
Girafarig with Twin Beam (Farigiraf)
Lickitung with Rollout (Lickilicky)
Mime Jr. with Mimic (Mr. Mime)
Piloswine with Ancient Power (Mamoswine)
Poipole with Dragon Pulse (Naganadel)
Qwilfish-Hisui with Barb Barrage (Overqwil)
Steenee with Stomp (Tsareena)
Tangela with Ancient Power (Tangrowth)
Yanma with Ancient Power (Yanmega)
There are also some Pokemon, like Primeape and Stantler, that evolve after using a certain move after a certain number of times. These are not included on my source, so they are not included here.
Note that in Legends: Arceus, Qwilfish-Hisui has to use Barb Barrage 20 times in the strong style to evolve into Overqwil.
Pokemon that evolve with certain moves (Bulbapedia)
