How to make Vegas Pro open AVI files
It seems that sometimes Vegas Pro's plugin that opens AVI files is corrupt or installed incorrectly (if this is the case, a reinstall may fix it). 

There is a simple fix that involves replacing the DLL file that controls AVI files.

The file is called aviplug.dll. The file path will be similar to: C:\Program Files\Sony\Vegas Pro 13.0\FileIO Plug-Ins\aviplug\


The fixed dll can be found at Mediafire.
Here's a YouTube walkthrough.