Just finished installing a "Intel(R) PRO/Wireless LAN 2100 3B Mini PCI Adapter" in my Webplayer - and against expectations it actually works!
The included documentation says "This product is intended for use only in Intel(R) Centrino(TM) mobile technology systems incorporating the Intel(R) Pentium(R) M processor..." but it works in the Webplayer and in an old PIII laptop as well.
They seem to go for around $40 in the US. I got mine for $45 (DKK 300) here in Denmark. Product overview is here: http://www.intel.com/products/mobiletechnology/docs/2100MPCI_ProdBrief.pdf
It only ships with Win2K and WinXP drivers, but I found Win98/ME drivers here: http://www-1.ibm.com/support/docview.wss?rs=0&q1=2100&uid=psg1MIGR-50565&loc=en_US&cs=utf-8&cc=us&lang=en
Just need to find a way to attach an antenna now... It has a "U.FL-R-SMT" connector (any good ideas, anyone?)