Hello, Scott here, no you don't need a car here in Feb. There are busses that run from Jackson to Teton Village about every 1/2 hr. Free shuttle busses around town. All with ski racks. Here is a link to the START bus info.
Parking at the village (JHMR) is an expensive nightmare. You are better off taking the bus.
Great time to visit Jackson Hole, the weather is usually great for the whole range of activities.