I play as Michael and removed a car from the garage by driving out (garage is now empty). When I try to order a car through the internet, it says "you have nowhere to store this vehicle". Why? My garage is empty now. Do I need a extra garage for a racecar?
In the answer to this question about garages: How does the garage system work in GTA V?
Taken from the manual: Player characters can own any vehicle in [GTA V] as long they have appropriate storage for it. To purchase a new car they must first own a garage ...
So yes, you must purchase another garage.