Dealers don't bid against each other becase if you send out requests for quote, one dealer won't know what the other dealer is quoting.
You have to decide the exact car you want first before you get to that stage.
So, start of with online research as to the type of vehicle you want, including doors, size, engine, price range.
Then go in person to the dealerships to drive a short list of cars you narrowed down.
Then if you want to collect quotes, you have to email dealers individually, and see what prices they come back with. You'll have better success if you specify the exact model, trim level, options, and color you want, as well as provide your info like name, phone email, and address. Anonymous requests for quote get thrown in the shredder. Also give the salesperson a shot who helpd you out on the test drive, and let them have a fair chance at earning your business since they invested some time with you.