Yes, a hdd can affect games, under certain conditions. If the game is fully loaded from the hdd, meaning its standalone and not web based, then an hdd can slow down the flow of data. This is more apparent in high end pc's that are used to uber detail games.
Being web based, LoL is more dependent net traffic, there's little that's actually in the ram, most is from the web servers combining your character with other players online, with only area details coming from your pc.
There's many reasons for a hipo pc to have low fps, including mobo drivers, cpu/gpu temps, VRM temps, driver conflicts, fps caps, v-sync, OC stability, eco settings, power settings, bios etc.
Pretty much unless your hdd is at capacity and really has no room for windows cache, or it's dieing a slow death with bad sectors, or is in dire need of optimizing/defrag, then there shouldn't be an issue in that respect.