I cannot find drivers that will work on a Sony Vaio PCG-FX240 running Windows ME - for the display adapter...can anyone help me?

You seem to have an ATI Mobility 1 graphics chip.

The best I could find was a Win98 file that may work also on Win ME

Try here (you might have to sign up for (free) membership.
http://www.video-drivers.com/drivers/52/52155.htm

here is the display driver you need, copy this link into your address bar. Good luck.

downloadcenter.intel.com/detail_desc.aspx?strstate=live&productid=178&dwnldid=4658&agr=n&lang=eng&prdmap=178

Intel 810 & 815 display driver for windows 9x/me

or just go to intel.com and search for the file win9x67, under intel graphic drivers.