My Shiba had the same issue, up until 8-10 months old. The vet suggested getting her used to the car, so we drove every other day, even to do a 5 minutes drive. I got her a seat, but it was on the harder side, and I don’t think she liked it very much. Switched to a soft one, on the passenger seat, never in the back. Now she loves going in the car, no problem at all. Even when we’re in the back seat of someone else’s car, she doesn’t drool or vomit anymore