There are two possible causes for the crash. The first is that Adobe Air crashes when it is brought back up. You can try updating it to version 3.6 here: http://get.adobe.com/air/
The second is that your air client files are a bit corrupt. You could try to replace them this way:
harddrive/Program Files/League of Legends/RADS/projects/...
delete the two folders 'lol_air_client' and 'lol_air_client_config_na'
Run the LoL Launcher, hit the cogwheel in the upper right hand corner.
Run a repair, let it finish, it'll download the deleted folders.
Run the game, test if you can play a game and load back up.
Lastly, if that doesn't work, it might be an actual issue with the game files. You could delete the game files and repair them, much like the above, only they would exist under the folder names of: 'lol_game_client' and 'lol_game_client_en_us'
The first step takes all of 5 minutes. The second and third can take hours depending on your internet connection.