is traveling with chicks bad for them?

They should be fine. I doubt you'll need the heat lamp if its a short drive... Is it? Maybe you can plug it into the car somehow but that'd be a hell of a lot of power...

 

If you're somewhere cold, warm the car up well and then put them in. If it's going to be a long drive and the travel box will fit on the front floorboard, you can put them there with the heat coming out the floorboard vents. You can also travel with hand warmer packs under a little shavings and they'll huddle around that heat source. I've made 5 hour trips with babies like this bringing them back from my buddies. In the spring once it gets warm, walmart and other stores have ridiculously low sales on those handwarmer packs, stock up then on them. Those little packs come in handy for lots of chick issues.