Android - How do I find out what wifi chipset my phone has?

Well without some specialized program, there's always dmesg.

Turn on wifi, then run dmesg in a terminal/adb shell/ConnectBot. Towards the end of the file you'll see debug statements about your wireless.

dmesg | grep -i lan can be useful if you have busybox installed.

I get lines like this:

TIWLAN: 1251 PG 1.2
tiwlan0: no IPv6 routers present

The first one isn't interesting on its own, but the second one screams "network adapter" due to driver naming convention for network adapters. "tiwlan0" is closer to OpenBSD's naming convention than Linux's (Linux native drivers would just call it wlan0) but...not unexpected.

A few web searches later - and yep, it's a TI wireless chip.

The sure fire method that will work for any phone is to crack open the case, write down the name and part number of the chips, and then research them individually. That can be pretty intensive though, not to mention that you might damage your phone in the process. Manufacturers typically do not publish the parts list for their products. The product spec sheet may contain the CPU type (a Qualcomm MSM7227 in the case of your phone). They may also list the GPU (Qualcomm Adreno 200 for your LG Optimus One P500). No guarantee on that though.

The folks over at isuppli.com teardown hardware and inventory the parts. Most of their reports you have to pay for, but some get published for free.

Most handsets use a chip from Texas Instruments, Broadcom, Royal Philips Electronics, or Atheros Communications. Using those chip maker names as a filter may help you to zero in on the particular chip in your handset.