Its really going to take you at least 8-9 hours to get from LA to the Grand Canyon. That's perfectly possible in one full day on the road, but I think it would be a bit much once you factor the time you're going to need to pick up the RV, get used to driving it, and stock up on supplies.
BTW, I'd recommend focusing on getting out of LA before worrying about groceries, just to reduce your chance of getting caught in traffic. I'd wait until you get someplace like Victorville or Barstow, when you'll probably want a break from the road anyway.
I suspect a good place to stop for the night would be around Kingman, or perhaps you might like to find a good spot along the Colorado River, I don't have any specific recommendations, but you should have plenty of options around Needles, Lake Havasu, or Laughlin.