Doesn't seem hard for a NEW game to include that. But riot is using a very old game engine. They would have to work their asses off coding that and then fixing all the bugs (Flash is bound to break).
The only real solution would be to add that chat as a overlay. This is actually the easiest option and if it breaks you have the option to 'Hide' the chat client or if it doesn't load you don't really lose anything and the game service wont crash with it.
Yea it won't happen. Not until the next season at minimum. Riot is massively behind DoTA2, and even fucking HoN in terms of features. They have made a dozen promises, Recording -- "Were working on it (Since S2). They can't even fix the Blue advantage over red and its season 5 now. Maybe they can figure it out but its on the bottom of their GIANT bucket list.