Often, motion sickness in puppies is caused by their nervousness rather than by the motion of the car itself. Because it's more about your puppy's emotional state than about the fact that they're in a moving vehicle, motion sickness can be overcome with patient and supportive training. Tackle motion sickness with the idea of comforting your puppy, and you should have a lifelong traveling companion in a few weeks.
Become good at reading your puppy's body language. The key to overcoming your puppy's anxiety about the car is to go slowly and to make sure she's relaxed before you move on to the next part of training, so be aware of what her relaxed state is, as well as what body language she uses when she first gets nervous.
Spend time with your puppy in your parked car, making it a positive experience for her. Bring something familiar and comforting out to the car, such as a blanket or toy that your puppy likes to snuggle with. Sit in the car with your puppy until she is completely relaxed and enjoying herself. Repeat until your puppy shows no nervousness whatsoever when you put her in the car.
Start the car, but keep it parked. Interact with your puppy in the same way you did in Step 2, until she has no anxiety about being in the car. Again, repeat until your puppy is completely relaxed about being in a running car.
Drive your car a few feet, such as out of your driveway, when your puppy is used to a parked car. Monitor her response carefully. If she has any nervousness, park the car again and wait until she's calm to move a few more feet. Once your puppy is relaxed when the car is moving for short distances, you can move on to the next step.
Go for a short drive with your puppy, stopping if she shows any signs of nervousness. The first time you drive with your puppy, plan on trips of 5 minutes or less. Increase the time by a couple of minutes when you repeat the step.
Take your puppy to a rewarding place in the car, such as to a favorite park or to visit a dog or puppy she likes to play with, once she's comfortable with short trips. The short trips should have shown her that there is nothing to be anxious about when in the car, and traveling to a rewarding place will show her that a car ride is actually something to be excited about.