AVI is listed as one of the supported formats by Sony Vegas Pro, why Vegas can’t import and open AVI files correctly? Well, the problem is mainly due to that Windows didn't come with the codecs that were used for those particular .AVI files(AVI is just a wrapper for a large variety of codecs.)... That’s why Sony Vegas Pro won't open it. So, you can try to download the codec packs to solve the issue.
And here we recommend you a better solution- transcoding AVI to Sony Vegas Pro more friendly video format like MPG. Brorsoft video converter is the right tool for you besides handbrake.