You might need a sled, not a cart. The "Just put it in the wood rack around back." Or "Stack it on the porch." Really, you want it on top of all those odd ball chunks in the half rotted falling down rack, behind the hedge, not to mention the 18" of snow on the tarp half covering it, the rest of the tarp froze in sculptured like forms.. Or... four thousand pounds stacked on the deck next to the door. Yes ma'am, I'll get right on that. Do you have a contractor on speed dial, or would you like me to recommend one?
The first time I delivered and stacked, I woke the guy up from his hangover. It took a while for him to come out. Then I thought he was looking for his check book in his car for ten minutes. Nope, he was looking for his cigarettes (or his brain). Then, he could not decide where to go with the wood, when I told him on the deck was not going to happen. I should have driven away after the forth cigarette, and still no sign of intelligence, ... but three of his neighbors walked over and ordered wood during the time I was waiting on him. Then a woman showed up from her morning jog, looked at him/looked at me, and said stack it next to the truck, and walked in the house.
To answer your question. I used a brick style wheel barrow, with a bit of effort in the snow.
I quickly learned to say no to stacking, making it very clear when they order, it will be thrown off the side or back of the truck. No stacking. No surprises.