This is my jam! I'll stick to birds because that's my specific area of special interest.
Bilgewater - Scarlet Macaw, obvious choice for both mascot and scout bird. Of course we all know what those look like.
Ionia - There are a lot of possible picks for this one, but my vote for a mascot would go to any of several species of bird of paradise, because they look borderline magical. As for scouting birds, I'd go with a White-throated Magpie-jay - it's almost like a fantasy version of an ordinary blue jay! And it's a corvid, which makes it quite an intelligent bird.
http://2.bp.blogspot.com/-JsmW2B0k19Y/T1gHIIw1NNI/AAAAAAAAB9E/08bm0v1P5-0/s1600/WhitethroatedMagpieJay2CostaJM.jpg
Noxus - While Red-winged Blackbirds seem to scream "NOXUS," I'd expect the empire to pick a bird of prey as its avian mascot, or as a scouting bird. Falconry is just one of the innumerable respectable military disciplines in Noxus, of course, so Quinn probably has quite a few counterparts on the other side. Check out a dark-morph Red-tailed Hawk:
http://www.schmoker.org/BirdPics/Photos/Raptors/RTHA_DkAdFly9.jpg
Side note: I've seen some fan art of Darius as a bird and he looks uncannily like one of these.
Demacia - They've probably already selected the azurite eagle (Valor's species, which isn't a thing IRL) for both tasks, so it's harder to pick a real-life bird for the job.
Shurima - You might think of ostriches when you think of the desert, but it's probably more feasible to select a bustard for a mascot:
https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Ardeotis_kori_Etosha.JPG/1280px-Ardeotis_kori_Etosha.JPG
Scouting birds would probably be hawks or falcons, just like in Noxus, but of course different types. Something like this Red-necked Falcon could do nicely:
https://upload.wikimedia.org/wikipedia/commons/0/0c/Red-Necked_Falcon.JPG
Freljord - None of the three tribes would go with a bird, I think. If they did? Sea eagle (this one is a Steller's Sea Eagle):
https://upload.wikimedia.org/wikipedia/commons/d/d2/Haliaeetus_pelagicus%2C_Rausu%2C_Hokkaido%2C_Japan.jpg
Piltover/Zaun and the areas uninhabited by living humans are harder to work out. Aside from those mysterious manta-like creatures in the Zaun artwork, I feel Piltover and Zaun would probably select mechanical birds as mascots.
And I'm gonna just agree with A Swarm of Koala about Bandle City!
Perhaps I lean too much toward birds of prey, but they're appropriately fierce mascots while also being intelligent enough to do scouting and perhaps even combat maneuvers with their handler - multipurpose!
Also, it's not just me: many major league sports teams in North America have chosen a bird of prey as a mascot: Hawks, Falcons, Seahawks, Raptors, Eagles...
