A quick scan of the forum's post has shows that the subject of vector format graphics has been touched on and generally it would seem that the best way forward would be to rasterize the 'ai' file to a png, tiff or psd file as these will support alpha transparency. In this case, it is probably best to save them at the largest size you will need for your project as bitmapped graphics don't enlarge very well - unless 'jaggies' are what you're after!
However, having said that... it is possible to use a vector 'ai' or 'eps' (version 8 or less) indirectly through a plugin like BorisFX Extruded EPS. Just make sure that the Extrusion Style in set to NONE, The Material Count is 1-Same on all sides and Front Material is set to Keep EPS Color.
I'm sure some of the more seasoned members of this forum have one or two tricks up their sleeves they could share ;))
.ai is the native Adobe Illustrator format - you should export files from Illustrator as 72dpi non-interlaced .png with alpha for best use in Vegas. Make sure they're large enough that you don't have to scale them up in Vegas.