Hi A. .. I may be late in responding to this, but my son has thrown up in his carseat many times. We simply take the entire seat out of the car, wash down the back seat in the car, take the cover off the carseat and wash it in the washer/dryer. If you set it on gentle cycle and low dryer setting, it is fine. My husband also hoses off the actual car seat (belts/padding and all), wipes it down with a towel and we let it dry for 24 hours near a heater in the house. This has worked wonders and the carseat and my car never smell like vomit. Febreze works wonders too!
I just noticed that your carseat cover is attached. Try putting some detergent in a spray bottle with water to dilute it. Spray the car seat down thoroughly, then take the carseat outside and spray off thoroughly with a hose or rinse in the bathtub. Blot dry with a towel to get the majority of the moisture out. Because the car seat cover does not come off, it will take a few days to completely dry but placing it near a heater (not too close) will help.
As for the carpet, an enzyme based product works great. We have one called "Kids & Pets" carpet and upholstery cleaner. Make sure you clean the area well with a carpet cleaner or soap and water (blot the area dry with a towel) and then spray the area with the enzyme cleaner. Ours came in a bottle with a screw top so we put it in a spray bottle. You want to saturate the area throughly but don't pour the product on to where you soak the carpet as it takes a lot longer to dry.
Hopefully this will help you! Good luck! =)