Get a 12V cooler instead.

You can’t. A passive cooler works by holding ice and insulating. Without ice (or frozen gel packs, dry ice, etc.), it’s just a box. Food will spoil within hours.

If the goal is zero ice, you need active cooling. A thermoelectric (Peltier) cooler runs off your car’s 12V port. They’re cheap but not very powerful — they’ll keep things cool, not cold, especially in hot weather. A compressor cooler (like a Dometic or ARB) is much better: it will freeze stuff and use less battery, but costs $500+.

For a weekend trip, frozen water bottles count as ice but at least you can drink the melt. If you truly want no ice at all, buy a proper 12V fridge. That’s the only real portable refrigerator sans ice.