We suggest crating the puppy or layering old towels three or four deep on the car seat or on your lap. It is not necessary for you to buy a small crate simply for transporting the puppy, however, as it’ll outgrow it quickly; but I strongly suggest it.
You can a small purse type soft sided carrier for the trip home and for the first week or so, when you first get the puppy home. When I travel over night with puppies and want a good night; I place the travel above my pillow on the bed. The first few nights the puppy has lost its litter mates to sleep with, so they will feel much more comfortable knowing you are not leaving them.
The car is a new environment and your puppy will most likely just sleep. If the puppy wakes up and wants to move around, it probably needs to potty. Use the collar/leash combo for stops during your drive. Some puppies will get car sick, so take a trash bag, paper towels, and wet wipes as a precaution. You can additionally layer the bottom of the crate with shredded paper (which I’ll happily provide upon request) to keep the puppy clean if it does get sick.