Oh wow, no doubt an ice car would lack heat and power. It could be towed or run with Flintstone power. I wonder if you knew the ice castle guys made a Dolorian from Back to the Future out of ice this year. It was spectacular! Whole castle behind. They used a chainsaw to carve it, I believe. Out of thick blocks of ice, so it was not the type of car you enter.
A couple years before this, there was an alien spaceship, two seater, rainbow color changer, fake controls carved out of ice. If you wanted you could put some wheels on that thing and run/roll it to your destination. It might help if the thing has a built in bowl so you can keep the bowl clean and brew kava out of it.
Also I am thinking no more than a mile transit. The point of going four hours early to the ice castle which is built in my town each winter, is to give the water time to freeze into a bowl thick enough to withstand kava brewing. The kava would end up being brewed at one unit above freezing temperature and absorbing ice so the ice you make the bowl of has to be sterile because it is for ingestion purpose. Place a catch underneath it, no need to waste kava if ice melts too quickly - that is why four hours, to get there, begin freezing a bowl, come and go and water it to thicken the ice, then use it for kava. Or more realistically to carve a bowl out of ice and spend hours doing that. The temperature has to be well below freezing day and night for weeks before the workers decide to create a castle anyways, and once it gets above freezing one time, they tear it down.