They could make it so that each week or month they release a "chapter" where you play as a champion through a lore segment of an ongoing story. Almost like how Blizzard does with their RTS games.
For example, the first chapter of the story mode could star Gangplank, pillaging the seas until he comes across something like a prototype ironclad vessel developed by Piltover and is defended by Jayce. Gangplank steals the ship, and the next mission has Jayce attempting to track the stolen machine, but has a run-in with Darius as he leads a battalion of soldiers, and from there the story basically kicks off as more champions get involved, the plot picks up, and the conflict escalates.