I just reported on this situation a few weeks ago after I was up there.
Marmots are known to be around the parking lot, and they will chew electrical wiring and chew rubber hoses, so they can do a lot of damage to a car in nothing flat. The park service has signs up warning visitors about marmot damage, and they have a standard form for you to fill out if your vehicle has been damaged by the furry critters. I don't know if that does you much good.
When I was there, I protected my car, then walked off with my backpack. About five minutes up the trail, I couldn't remember whether I had locked the car or not, so I set my pack down and dashed down the trail. The darn marmots were already surrounding the car and plotting their attack. So, this is not a fantasy.
There are two general methods of protection. One is chicken wire, and you would need enough to surround the bottom two feet of your vehicle. I have heard rumor that there is a store up there that will rent you a length of chicken wire, and the rental fee is about the same as what it would cost you to purchase the chicken wire at home. I can't confirm that.
The other method which I chose uses a simple woven blue tarp. It must be longer and wider than your vehicle by about five or six feet in each dimension. You stretch it out on the dusty parking lot, then drive your vehicle over it. Then you pull up the sides and corners and tie them all together with parachute cord or something. Ideally the tarp will cover at least the bottom two feet of your vehicle. I managed to get my tarp to cover the wheel wells.
I got back about three days later, and the tarp was intact, so the car was intact.