I'm not entirely sure. Riot did have an achievements tab in the summoner profile for a while they never seemed to get around to fully implementing.
Extra Crediz did do an episode talking about a lot of the problems involved with such a system though, approaching the problem from an angle around making PvP games have a stronger storyline.
In general, it seems most story-elements, achievements, and side goals will frequently run into issues where players will intentionally do mildly to massively detrimental things to the PvP experience in pursuit of such rewards.
And the way LoL is currently balanced and set-up, the only real factor is a champion's capacity to topple the opposition's Nexus.
I'm not saying it shouldn't be done - it would be really cool if Riot implemented various achievements that help shed some light on the more archaic play-patterns of champions in LoL. It could be a powerful tool to allow players to figure out you really don't want to duel with some champions and you do want to 1v1 to win a game with others, and it could point out where less-flashy plays are having a bigger impact.
It's just - you'd want to word such achievements very carefully. Like, Land a 5-man Morganna stun while 3 of your teammates are within 600 units. Over just a 5-man stun.
Worded in this way, you encourage players to perform the optimally strategy of utilizing powerful team-fight tools in a team-fight setting where your allies don't need to burn flash to fallow up on a game-changing play, without running into issues of trolls or just mislead players thinking they made a good play by simply suiciding into the enemy team.
Achievements could have a lot of power in incentivising warding, team-fighting as team-reliant champions, or just giving hints on optimal strategies.