1.  Beach Towel (home made)
2.  Snow Scoop 
3.  Lift-N-Aid

SCOOP

Available at  Improvements Catalog   item # 135863   56" long  28" wide Clean Air Gardening
I would add 2 more wheels for stability, and a rope at the handholds. Roll to vehicle, put one end into car, slide unit with dog on it, onto the seat. Secure with seat belts...and off to the Vet !