We've got a little Jack Russell girl who we got from a dog shelter, and when we got her she wouldn't even go near the car. As well as not being house trained, no basic commands and no name. Pretty hopeless little case!
When I'd drive her anywhere she would pant, drool, whine, shake, generally all of the worried signs apart from vomiting. I decided from the start to just try and ignore all of the behaviour that wasn't appropriate (as I have done with her normal training) and use positive reinforcement like vocal praise to reward her when she's doing something I want, like sitting in the footwell and not making a fuss (she's not motivated by food at all so praise was all I could use!) or sitting on the mat we got her for the car.
It wasn't a quick solution, took about 6 weeks of daily short journeys, very slowly increasing the distance, but it has worked and she now can't wait to get in the car. I sometimes feed her in the car too which she quite likes. Or we'll randomly take one of her toys in there so there's something familiar.
Tried to just keep myself calm and remember how reactive to us they can be; i.e. If we're fussing over them being in the car, they're more likely to think something worth panicking about is happening!
Best of luck, I'm sure you'll get there eventually. It's very frustrating!