You've reopened Pandora's box. This topic has come up more than once on this forum with passionate arguments on both sides.
I'm on the side of not using propane while in motion. It's illegal in tunnels and at gas stations and on some bridges. It's illegal because open flame is bad. People will tell you they've done it for years with no problem, or that propane, gas fumes, dust, and other combustibles only ignite at very specific concentrations. Unfortunately, the laws regarding open flames or running propane appliances while in motion were written for a reason. Despite safeguards at gas stations and elsewhere, accidents happen; you don't want to be the spark (or flame) that turns an accident into a tragidy.
From a strictly legal standpoint, are you really going to pull over before the gas station (tunnel, bridge, etc), shut off your fridge and propane, pull in and fuel your vehicle, then pull out, pull over, then turn on your propane and fridge and continue your journey?
If you're transporting perishable foods, get them cold before departure, monitor the temp en route, and learn how long foods can last above 42 degrees before needing to be tossed or reprocessed. If you just want a cold beer at your destination, buy a little cooler and some ice.
2011 Jayfeather X23B
2001 Ford Expedition Eddie Bauer