This is a possibility, especially since puppies are not used to the sensations
of travel. If you are taking your pup home for the first time, it is also
likely to be distressed at the separation from its littermates, and whining
could exacerbate the problem.
If your puppy vomits, stop the care and replace the bedding at the earliest
opportunity, so that its coat doesn't become soiled.
To overcome motion sickness, take your puppy out regularly for short drives,
and it should grown out of the tendency. It is a good idea, in any case, to
accustom dogs to travel from an early age, so they will not be nervous about
car journeys in later life. If traveling continues to upset your dog, consult
your veterinarian and ask him or her to provide suitable medication to calm
it when necessary.
There are also herbal products available for this purpose.