Of course.
You could have one without the other. Hell, irl examples of this would be sports and literature.
For video games, at least the ones that are developed past the point of being Newgrounds-level flash games, have some degree of story to it. Some stories to those games are paper-thin, while others are actually pretty hefty considering the type of game it is.
For a game as big as League of Legends, and for champions to have introductory cinematics (Bard and Kalista, I'm looking at you), I expect a level of story to justify those things.
I guess I can look at other games that can technically say they're in the same page: Fighting games.
Gameplay-wise, there's little exposition given to characters like Ryu and Ken when they fight in Street Fighter. Yet, I can't think of a gameplay trailer that focused on Ryu and his journey to improve as a fighter and control his inner rage that makes me care more about him as a character. There's movies, yes, there's the cinematics in arcade mode, yes, but it doesn't stop the game to focus on that character's story to do so (at least, i think so).
Of course, given that lore isn't integrated into actual gameplay, but rather provided "in the background" so to speak, it works for those games. Same can be said here, except the background stuff is lacking.