I'm developing a webapp and it would be awesome if there's a way to find (or extract) the icons (i.e.: turrets, inhibitors, pings, champion circles, nexus, baron, etc.)
All of the game assets are packed into RAF files that can be extracted using this tool:
They tend to be in this folder: C:\Riot Games\League of Legends\RADS\projects\lol_game_client\filearchives
There you will find a ton of folders (one for each patch). Content generally resides in the folder of the patch on which it came, or possibly even the patch you installed the game on. Just sort the folders by content size and start from the largest, you will eventually find the right RAF file.
Once you find the files they could be any number of formats. Hopefully they will be png and you will be good to go but some may be dds or something like that. A quick web search of the extension should point you in the direction of a tool that can open that type.
Google or Bing Images + Photoshop or GIMP and construct the images on your own.
Some sites like lolking or lolpro have a number of the assets already extracted or chopped up, you can always grab them from these sites.
Chances are a combination of the above will get you everything you need asset wise. If you need any game statistics RIOT has made them available through this API: https://developer.riotgames.com/