You need to either convert the video you have, or export a suitable video from Vegas.
What format is your video right now?
You should be able to upload a QuickTime .mov, an MP2 or MP4, or a Windows AVI or WMV file to YouTube no problem. (Though they might still have that 10 minute limitation).
Once you have your video ready on your computer, you can upload it via your YouTube account. (After it has uploaded, YouTube converts it automatically to their special codec. Sometimes that can take up to an hour, depending on length and original format). Once it's up there, you can copy the EMBED code (available on your video details, at YT) and paste that into your actual website. (Don't forget to enable remote embedding, in your YT options).
It's pretty simple, once you've done it once.