You said the monitor works fine with other machines, On the one your having problems with, does it show POST (stuff before it loads windows,) or does it just not work at all? If its showing the text then theres an issue between your monitor and Vista.
If It does not show anything and you want to get a new graphics card, First see if theres a waranty or something on your machine. If you do not have any warrenty, a new Graphics card may be in order. These can be a bit expensive however, so you should wait and see if anyone here comes up with another solution. Your machine appears to have an open PCI slot so you need a card labeled PCI Express 16. This page has some examples (http://www.tigerdirect.com/applications/category/category_slc.asp?CatId=1558)
Your current graphics system has 128 mb shared memory, and generaly speaking, the more memory the better. If you do anything that requires a good video card, you should be looking at atleast 512mb. 256 mb is pretty solid if you're just looking for a standard card, and these tend to be about $50 USD.