No, you don't need to jack up the car to change out the valve core. Make sure to have your replacement core at the ready. When you pull one core out, insert the other core. Some of the air in the tire will come out, of course, but you won't cause any damage to the tire. Stick your new core into the valve stem and tighten it. You may want to wear some nitrile or latex gloves while you are doing this. The air coming out of the tire will be cold, colder in fact than the outside air temperature. This could cause spots of frost bite on your finger tips if exposed to the escaping air for too long. The gloves will protect them well enough while you do your work.
You do need to know if the vehicle is equipped with a Tire Pressure Monitoring System (TPMS), though. Some of these have different valve cores than the normal cores, so replace with the right kind.