Can I trade in a car that doesn't run?

Yes, many dealers and instant-offer buyers will still make an offer on a non-running car, though it will be much lower and they may need to tow it. Be upfront about the problem, because the offer will be adjusted after inspection anyway. If the car has little value, compare dealer offers with junk-car or salvage buyers, and check whether repairing it would raise its value by more than the repair cost. Make sure you have the title ready.