Scarecrow: Not commonly a living being, but in one specific case, a scarecrow was brought to life.
Champions: Fiddlesticks, receives quite a bit of attention, mage

Rats: Common sewer rats brought to life by excess chemicals and necromancy. Not given resepect in Valoran
Champions: Twitch is the first representative of the Rat-People, Doesn't receive a ton of attention, Stealth/DPS.

Undead: Any being previously dead, reanimated (Likely by necromancy)
Champions: Sion, Karthus, Urgot (Also a cyborg) These three are all vastly different types.

Wolves: Naturally born wolves are non-sentient beings who roam forests
Champions: Warwick, previously a human, Warwick was transformed into a wolf to reflect his evil mind and soul. There are also small packs of natural-born wolves which act as neutral monsters.

Robots: A man-made race consisting of machinery and oil as opposed to flesh and Blood
Champions: Blitzcrank, receives a fair amount of attention, Urgot is often considered a robot or cyborg.

Manatees: A rather proud race, their whereabouts unkown
Champions: Urf (R.I.P.) Urf was to be the first champion from the race of manatees. Unfourtunately, he was killed by Warwick before he could be formally accepted into the League

Mordekaiser is possibly undead (ghost in the shell type of deal).

Evelynn is possibly undead (vampirism theory).

Yorick is most definitely undead.

Does Baron count as an non-sentient being? I think he'd count as a Wyrm.

Great list.

Don't Forget To Add Skarner When He Comes Out and why are lizards not in non-sentient beings?

Added some stuff, and lizards have weapons and armor.

It saddens me to see no Dwarves...

Gragas is a dwarf, or so i believe