8 species have Run Away as a hidden ability, 22 as a normal one

Effect, in PokéAPI's words

“Ensures success fleeing from wild battles.”

As a hidden ability

Species	Slot	Note
Caterpie	Hidden	Since Generation V
Weedle	Hidden	Since Generation V
Oddish	Hidden	Since Generation V
Venonat	Hidden	Since Generation V
Wurmple	Hidden	Since Generation V
Nincada	Hidden	Since Generation V
Kricketot	Hidden	Since Generation V
Lillipup	Hidden

As a normal ability

Species	Slot	Note
Rattata	Normal
Raticate	Normal
Ponyta	Normal
Rapidash	Normal
Doduo	Normal
Dodrio	Normal
Eevee	Normal
Sentret	Normal
Furret	Normal
Aipom	Normal
Dunsparce	Normal
Snubbull	Normal
Poochyena	Normal
Pachirisu	Normal
Buneary	Normal
Patrat	Normal
Nickit	Normal
Thievul	Normal
Wooloo	Normal
Tandemaus	Normal
Maschiff	Normal
Dudunsparce	Normal

Elsewhere in PokéAPI

6 more holders without a page here

Alternate forms and other species that PokéAPI lists with Run Away.

Eevee Starter, Ponyta Galar, Rapidash Galar, Eevee Gmax, Dudunsparce Three Segment, Gimmighoul Roaming