My Eevee is level 21 (and she is so darn precious! :D) (<--- idk) and doesn't have a Fairy move so for now I can't evolve it ;-; When is the next time it will learn one?
If bred, Eevee will learn Baby Doll Eyes at level 9.
If caught at level 19, it will already know Baby Doll Eyes.
If caught at a higher level, it will not have Baby Doll Eyes, but will learn Charm at level 29. You can also use the Move Relearner.
Eevee learns Charm at lvl 29 or by breeding
Hope I helped! :D