this only took a few minutes of looking at a prize car list:
-Chrysler Crossfire
-Subaru 360
-any truck
-Toyota Vitz/Yaris/Echo
-any Isuzu
-Mazda Miata/MX-5/Roadster
-any(?) Aston Martin
-Opel Speedster/Vauxhall VX220
-any TVR
-Daihatsu Midget
-Daihatsu Copen
-Honda Civic Type R (for both the Civic race and the type R race)
-Mitsubishi Mirage
-any Nissan kei car (for the March brothers race, there's a bunch of cars that I don't remember the names of that qualify. I don't think you can win any of them)
-any Suzuki kei car
-Toyota Altezza/Lexus IS
-Saleen S7
-any Shelby (IIRC, they all qualify for the Cobra cup)
-Audi TT
-Audi A3
-either BMW 1 series
-Lotus Elise
-Mercedies-Benz SL class (you can win both MB races with it)
-either MG
-Peugeot 206
-Renault Clio
-Renault Megane
-Triumph Spitfire
-VW Beetle
-any VW GTI
There may be some races that a prize car can enter, but probably not win, that I missed. There may also be some peculiaritys of qualification I didn't see, but this gives you a good idea of how impossible it is to get 100% without buying anything.