It's possible to retrofit an internal wireless card into the Thinkpad A31p. Various instructions on the web. This looks fairly sane:
The big headache is where to put the antennas. Some models came with internal antennas, others did not. The easiest, but worst performer is under the keyboard. Gluing them into the LCD display is best, but takes much more work. One antenna, connected to the MAIN port, is sufficient, with the other antenna in the base unit. I've done two of these. The problem was that the IBM BIOS checks if the wi-fi card has been blessed by the FCC for operation in the laptop. As I vaguely recall, the A31p checked the card, while the A31 did not. So, I had to buy Intel 2200BG cards made for IBM in eBay. About $20/ea.
Well, if it needs an antenna connector, your selection is limited.
For PCMCIA/CARDBUS, I like Buffalo Tech WLI-CB-G54S. The problem is that Buffalo doesn't sell it any more. None on eBay and only a few for sale on Amazon at high prices. Oh well.
2nd best is something from Ubiquiti.
The catch here is that there's no internal antenna so you MUST use an external antenna. +24dBm tx output (200 mw) on 2.4GHz.
I use an A31p running Ubuntu for site surveys. The 1024x768 screen makes it useless for anything more elaborate. When I plug in a CARDBUS adapter, and then throw it into my laptop back, I invariably end up breaking off the antenna section. I've trashed two Netgear something cards this way. If you're doing CARDBUS, be sure to unplug the card before carrying.