They currently have a Windows based client to upload the files whenever the user asks it to, however I felt that this step was not really necessary. I hunted around to see if someone else had automated the process but unfortunately not, so I set about doing it myself. A couple of days later and Jax Uploader was finished!
Built in C#, it will watch for you to close LoL before attempting to upload the log files (since they are written to the disk in exclusive mode until then). It's very simple and does exactly what it says on the tin. It's overall quite a nice small project as its needs a few key features such as process scanning and holding, registry reading/writing and storing user settings correctly. Very good for getting me back into C# at least.
It's missing auto updating (or even knowing that it could be updated) right now, but that's something i'm going to be working on soon. Until then you can download it here...
Jax Uploader (v1.0.0.6)
It requires the .NET framework version 4, so make sure you've got that before trying to run it (otherwise it'll probably explode in your face and you'll feel very stupid indeed!)