Someone posted a fix on Reddit.
This problem can be solved yourself. Here is how to FIX it:
Go to League of "Legends\RADS\solutions\lol_game_client_sln\releases\0.0.1.84\deploy\DATA\menu\hud" folder location.
Open up the file that has your screen resolution in the name. Mine is hud1680x1050.ini.
Find [PlayerHudChat] - Add two lines, one for XPos= and one for YPos=
For mine I have it as the following:
[PlayerHudChat]
Scale=1.0000
YPos=-93.000
XPos=90.000
It will take a little putzing around to get it just right, but it will stick. I recommend that you spectate a game, and then exit the game, but leave the 'Reconnect/Quit' window up and go modify the values for XPos and YPos.
I then recommend that you save a copy of this file somewhere, so when there is an update to LoL, and it hasn't been fixed yet, you can easily copy/paste it and have your chat settings set right away without having to dick around with it again.