If we're going solely by lore, then the weakest champion in the game is Rumble.
It's mainly because he relies on a suit that barely holds itself together and without it, he's powerless. You'd probably laugh or be incredibly confused if he announced that he was going to fight against trained soldiers, mages armed with forbidden power, and unknown horrors with that thing. If that was to represent what yordle technology can really do, then you'd probably run away really fast from anything made by a yordle in fear of it blowing up in your face.
The only reason he was even allowed in actually was because he hated Heimer so much that when his suit got one-shotted, he was willing to take on Heimerdinger piloting a mecha ten times Tristy's size with his bare hands.
Then I bust out Sudowoodo, and you're like "lol grass" so you try fire BUT IT DOESN'T WORK BECAUSE SUDOWOODO IS A ****ING ROCK!
You can't stipulate that its Rumble because you separated him from his mech. That's like saying Sona without her etwal, or Draven without his ego. It's an integral part of the champion.
As for who I think is weakest, its probably some ADC. They generally are just standard people who happen to be reasonably skilled and have some supplementary gadgets. Cait, Draven, Ashe, Fiora, they all could easily take on any standard person, but matching them against gods, warrior kings, eldritch horrors, and abominations just doesn't seem fair.
The guy who thinks Karma is a viable champion. Hero's aren't made. They're cornered.