I recently bought a PSP car adaptor that functions perfectly with axim.
Just tested it on a trip, used my Axim x51V with a Garmin GPS receiver.
It charges while the Axim is on.
The downside? quite expensive, $25 at Bestbuy.com
Sony - PSP Car Adapter - 98528
Couldn't find a cheaper one that does the same trick.
There are some older articles all over the internet on a PSP charger found at Walmart that was <10$, but I am in NY and we have no Walmarts, and couldn't find the described charger at walmart.com
good luck!