To the best of my knowledge the only way is to open it up and look at it... sorry..
Unless you have some documentation that came with it.

at boot the computer lists installed devices. Try to press pause at that screen. It should list your audio card if it's plug and play. Put the numbers into google it might find something. I just tested it and if you cant catch it with pause try to put a unbootable floppy in your floopy drive and set the comp to boot from it. Then it will pause with an error e.g. non bootable disk error press any key...
 
PCI device listing ...
 
VENDOR ID DEVICE ID DEVICE CLASS
1106 0571 IDE CONTROLLER
10B7 9055 NETWORK ...
10DE 0111 DISPLAY
 
Then go here http://inf33-www.informatik.unibw-muenchen.de/infothek/computer/tutorial/eprmhtml/eprmx/12529.htm
 
There are more lists like this if you put ("vendor ID" list) into google.
From that you can see that
1106 is VIA Technologies
10B7 is 3com
10DE is nVidia
 
Those are chips manufacturers not the actual manufacturers, but you can try and get a generic driver.
 
 
BTW this won't work with old isa cards & first check if it's not an onboard audio.
 
If this doesn't yield any results try a program like sisoft sandra under AGP PCI card busses information you might find what.
 
Post back if you need more help.

Get AIDA32 Enterprise Edition 3.88 from
 
http://www.aida32.hu/download/aida32ee_388.exe
 
Install and run it and a report will be produced showing you the details of your hardware and software.