By any chance did you keep all the assets in their own directory specific for this project? If not, you should have, and please try to do so for future projects. It makes this task so much easier. If you have done so, simply copy that entire directory to the other computer. Open up the .veg file. Vegas will most likely complain that it can't find the first file on the timeline. There's an option to specify replacement files. Browse to the directory where they are stored on the new computer and select the file. Vegas is smart enough to find the rest of them there.

If you've got files stuck all over everywhere then there is another method that is very messy. Save the project and select the checkbox that says 'save media with project'. Choose a new empty directory. Vegas will make copies of all the assets in that directory along with a new copy of the .veg file. It's messy because Vegas splits the files up into lots of little pieces with new names, and often separates the audio into separate files. And, you will probably still have to do the search step mentioned above anyway.