Firstly some good news.
Your salesman was quite correct, the TRV22E is a DV-IN enabled camcorder. The instruction for recording to this camera using i-LINK (Sony speak for firewire) is discussed on page 123 of the UK manual.
Secondly, to successfully record from a computer it is essential that the video produced by your software is set to DV Encoder Type 1. This is perfectly possible with ULead Video Studio v7. I don't use Vegas myself but would be extremely surprised if it could not encode DV Type 1.
Whilst I haven't exported from ULead to the TRV22, I have to the TRV33 (TRV22 big sister). So I suspect that the problem is related to your project settings when you render the video. As far as ULead is concerned it is worth checking the options button when you create the video. I noticed on V8 that it defaults to type 2 instead of type 1.
Hope that helps somewhat. Post back and let us know how you get on.
COMPAQ something or other with 500Gb Boot Disk, 1Tb external Firewire Disk, 4Gb Memory and a super duper sound card, 19" TFT Monitor etc etc etc
Sony TRV33e : Sony TRV310e : Canon XM2
Sony Vegas Studio editions to v11
Remember, there is always more to learn than there is to teach.