Yes a dog can scratch a vehicles clear-coat. I've seen a dog scratch deeply into the clear-coat, but never down into the paint itself. I'm not saying they can't get to the paint, but it would be tough.
Chances are that only the top layer of the clear-coat is scratched. A simple compound and wax should do the trick of restoring the finish to its original state.
Try the compound and wax and if that don't do the trick, follow through with the info that flyingron has provided.

