It's worth noting that you must actually buy all of the vehicles through the computer in the Batcave or Arkham. While each of the vehicle levels gives you each of the story mode playable vehicles for free, there are others made available to buy by completing other levels.
You will need to spend a total of 752,000 studs to buy the remaining vehicles.
Vehicle unlocks by level:
Hero:
1-3
Batmobile
Batcycle
Police Car (10,000)
Police Bike (11,000)
Police Van (13,000)
The Joker's Van (60,000)
2-2
Batboat
Robin's Watercraft
Robin's Submarine (30,000)
Penguin Goon Submarine (15,000)
Harbor Helicopter (16,000)
3-1
Mad Hatter's Steamboat (23,000)
Mad Hatter's Glider (18,000)
3-3
Batwing
Batcopter
Bruce Wayne's Private Jet (15,000)
3-5
Bat-Tank (200,000) (Assuming you have completed all 3 hero stories)
Villian:
1-2
Mr. Freeze's Kart (25,000)
Mr. Freeze's Iceberg (23,000)
1-5
Two-Face's Armored Truck (24,000)
Riddler's Jet (17,000)
2-2
Catwoman's Motorcycle (23,000)
2-3
Penguin's Submarine
Killer Croc's Swamp Rider
Police Watercraft (12,000)
Police Boat (13,000)
3-1
Harley Quinn's Hammer Truck (80,000)
3-2
The Joker's Helicopter
The Scarecrow's Biplane
Police Helicopter (14,000)
Goon Helicopter (11,000)
3-4
Garbage Truck (99,000)