I have a Palm TX and would like to use it on our office WiFi LAN. 
Unfortunately, it is secured with WPA and Protected EAP (PEAP).  It
looks like the Palm WiFi client only does Personal WPA (using a shared
key).  How can I, including installing a 3rd party client, get the
Palm TX to work wirelessly (using WiFi) on this existing network
without opening a hole in the WLAN security?

Unfortunately, Microsoft had a hand in PEAP development,
and appears to be the supplier for implementations,
so it's unlikely that there will be a Palm client.

Too bad, you've been locked out.

See http://www.intel.com/support/wireless/wlan/sb/cs-008413.htm

However, Funk (who has a WinCE client) has been acquired by juniper
networks (www.juniper.net) and *might* have a different point of view.

Good luck!