I wonder if I could say something like:
"I feel nauseous."
"I got carsick."
when I feel sick and going to vomit due to a motion in a car. Does the sentences above sound natural?
Both sound natural but are referring to different points in time in your unfortunate situation.
"I feel nauseous" is present tense (i.e. right now you feel sick), the better of the two choices for saying you might vomit soon.
"I got carsick" is past tense (i.e. at some point you became sick) and will most likely be what you say after you have already vomited.
As Weather Vane in the comments pointed out, You can also throw in the immediate future tense by saying "I am going to be sick," which is the best option for informing people that things are not going well for you and are about to be worse.