Use your new Vista computer to go to the E-Machines website and download the drivers you need.  Put them on a flash drive or cd and then take them to the E-Machine and solve your problems.  You can download SP2 from Microsoft if you don't have it on the E-Machine or perhaps you can still order it from Microsoft for free.

You might also go to add remove programs and in the windows section add the necessay clients.  

I went to WalMart and for I believe $14.95 or so purchased a NIC card that came with drivers.   I see that you can get it cheaper online at http://www.walmart.com/catalog/product.do?product_id=2067722 , nice thing about WalMart is if it doesn't work or you decide you don't need it you can take it back.

Last thought, you could download Ubuntu from  http://www.ubuntu.com/GetUbuntu/download , burn it to a cd and use it to run your E-Machine  IF it has 256MB of ram or more.  You put the CD in, boot up on the CD and it comes with FireFox and a lot of other nifty software.  Plus you have little chance of getting a virus or other junk.  Lastly it runs faster.  If you like it you can install it by clicking on the INSTALL button.  The download will be an ISO so you will need to burn the ISO using something that says burn image, not just copy the ISO to a cd.