[Help] Dog drools and vomits in car

A bit of backstory. We adopted our dog (a whippet boxer mix) 4 months ago and she has had terrible car anxiety since we brought her home. Before we get into the car she starts drooling really bad, but she willingly will jump up in the car. She likes to curl up and lay down and sometimes she will stand up and look around. She will also stand up and puke in the car, although she tries and holds it in.

I have read a lot of posts on here that say start slow by walking her around the car, then moving up to sitting in the car with it off, and that it could take weeks. My only concern is she goes to dog care 2x a week and I don't want to cut that out of her life because she loves it so much.

Is there a way we can work on her anxiety and still take her to daycare/training?