I have officially given up.
I disabled all encryption in the access point and got the exact same results I had when I was using WPA. I tried with both the native Linux driver and ndiswrapper with the XP driver.
Either way, the network is seen and appears in ifconfig and the details in iwconfig are correct. It shows up in wifi-radar and even says it's connected, but I can't ping my router or the access point and I can't connect to the Net. When I try to ping, I get "Destination Host Unreachable."
I've checked everything many times--dmesg, system logs. Nothing looks amiss. But it doesn't work. I can't ping anything or go online.
So I'm done. I've spent many, many hours on this with no results. I'm pretty much disgusted. It shouldn't be so hard.
The wireless network is the number one problem, but close behind it is the inability to hear any sound. The problem is frequently reported on the Web. I've tried all suggested solutions, but nothing works. I've given up on that, too. No more time to fool with it.
So I'll continue using VL in my VirtualBox virtual machine under Vista. I have enough computer power and RAM that there is no perceptible performance hit. Best of all, everything works.
I've discovered that I like the virtual machine so well that I'm going to put it on my XP computers, too--although they have VL partitions that are working perfectly. With the virtual machine, everything is at my fingertips. I can run Linux most of the time, but if I need a Windows program (as I sometimes do), I don't have to fool with Wine or leave Linux and boot to Windows.
I'll leave the Linux partitions as they are and when 64-bit VectorLinux gets farther along, I'll install it and beta test it. Maybe a newer version will work with this hardware.
Thanks to everyone who tried to help.