You can find logs of events under Harddrive:RiotGames:LeagueOfLegends:Logs:Gamelogs (pick one) and skim each one.
Steam has a clean folder listing all nicknames under loginusers.vdf
Does Riotlauncher have an equal?
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Sign up to join this communityRiot's client ("Riot Client") doesn't seem to persist nicknames/usernames anywhere in any of its files. If it does, it's nowhere that's publicly-visible (i.e. not obtainable without having a tool to read the files they store.) So the clear answer here is no, there's no folder with all nicknames.
You can find the summoner name used on your computer by parsing through the Gamelogs, as you mentioned. If you open up (League of Legends folder)/Logs/GameLogs
and grab one of the r3dlogs on a particular date you played, you can open the log file and find the line that includes "LOCAL". It'll be stated next to the summoner name in the list of players.
Here's the particular line that indicates that my name (Spevacus) along with "LOCAL" in a log of a game that I played a long time ago:
000001.258| ALWAYS| ROST| CONNECTION READY | TeamChaos 1) 'Spevacus' **LOCAL** - Champion(Gragas) SkinID(3) TeamBuilderRole(NONE) PUUID(c4038cb4-ca42-56b4-9c7c-d6f4e547c07a) ConnectionState(Connected)
You could technically use this to get the summoner names of all users who play on your computer so long as you have access to the file folders for that user. For most people, that's likely not enough to, say, recover an account, though, as these are summoner names, not account names.