You only have to get into the car for it to count, so you don't have to drive it anywhere. There are 36 cars in the carpedia in the game, but there are more than that in the main game, each counts.
Also there isn't a list as comprehensive as mine in any solution thus far to my knowledge.
List of every car that shows up in the entire main game with NO DLC in Alphabetical order.
Ascot Bailey S200 Chap 11-15
Berkley Kingfisher Chap 9-15
Culver Empire Chap 2-15
--Police Special Chap 2-5
Dezilia 410 Grand American - I've read this is in the main game but in almost everywhere it also states its NOT in the main game, I personally have not seen it, I don't think it's there.
GAI 353 Military Truck Chap 2-5
Hank B Chap 7-15
--Fuel Tank Chap 7-15
Houston Wasp Chap 8-15
ISW 508 Chap 11-15
Jefferson Provincial Chap 2-8
Lassiter Series 69 Chap 2-10
Lassiter Series 75 Hollywood Chap 7-15
Milk Truck Chap 7-15
Parry City Bus Chap 2-15
Potomac Indian (Chap 8's you should store as it's the best) Chap 7-15
Quicksilver Windsor Chap 9-15
--Taxi Chap 7-15
Shubert 38 Chap 2-15
--Hearse Chap 2-15
--Panel Truck Chap 2-15
--Taxi Chap 2-5
Shubert Armored Van Chap 7-15
Shubert Beverly Chap 8-15
Shubert Frigate Chap 7-15
Shubert Pickup Chap 2-15
Shubert Hot Rod Chap 15 (rare spawn)
Shubert Series AB Chap 2-5
Shubert Snow Plow Chap 2-5
Shubert Truck (Truck, Flatbed, Covered - not sure if all 3 count as just 1 or if it counts for 3 different cars) Chap 2-15
Smith 34 Hot Rod Chap 8 ONLY (this may not count at all)
Smith Coupe Chap 2-5
Smith Custom 200 Chap 9-15
--Police Special Chap 7-15
Smith Deluxe Station Wagon Chap 7-15
Smith Mainline Chap 7-15
Smith Thunderbolt Chap 11-15
Smith Truck Chap 2-5
Smith V8 Chap 2-8
Walker Rocket Chap 11-15
Walter Coupe Chap 2-10
Walter Hot Rod Chap 15 (rare spawn)
Or if you prefer by chapters they show up in:
Culver Empire Police Special Chap 2-5
GAI 353 Military Truck Chap 2-5
Shubert Taxi Chap 2-5
Shubert Series AB Chap 2-5
Shubert Snow Plow Chap 2-5
Smith Coupe Chap 2-5
Smith Truck Chap 2-5
Jefferson Provincial Chap 2-8
Smith V8 Chap 2-8
Lassiter Series 69 Chap 2-10
Walter Coupe Chap 2-10
Culver Empire Chap 2-15
Parry City Bus Chap 2-15
Shubert 38 Chap 2-15
Shubert Hearse Chap 2-15
Shubert Panel Truck Chap 2-15
Shubert Pickup Chap 2-15
Shubert Truck (Truck, Flatbed, Covered - not sure if all 3 count as just 1 or if it counts for 3 different cars) Chap 2-15
Hank B Chap 7-15
Hank B Fuel Tank Chap 7-15
Lassiter Series 75 Hollywood Chap 7-15
Milk Truck Chap 7-15
Potomac Indian (In Chap 8 you can get one, it's the best one, and you can store it, so do it) Chap 7-15
Quicksilver Windsor Taxi Chap 7-15
Shubert Armored Van Chap 7-15
Shubert Frigate Chap 7-15
Smith Custom 200 Police Special Chap 7-15
Smith Deluxe Station Wagon Chap 7-15
Smith Mainline Chap 7-15
Smith 34 Hot Rod Chap 8 ONLY (this may not count at all)
Houston Wasp Chap 8-15
Shubert Beverly Chap 8-15
Berkley Kingfisher Chap 9-15
Quicksilver Windsor Chap 9-15
Smith Custom 200 Chap 9-15
Ascot Bailey S200 Chap 11-15
ISW 508 Chap 11-15
Smith Thunderbolt Chap 11-15
Walker Rocket Chap 11-15
Shubert Hot Rod Chap 15 (rare spawn)
Walter Hot Rod Chap 15 (rare spawn)
Unknown Chapters - Dezilia 410 Grand American - I've read this is in the main game but in almost everywhere it also states its NOT in the main game, I personally have not seen it, I don't think it's there.
I created this because I didn't see a full list of every car and it's chapters anywhere so I made one with the use of the Mafia wikia.
If you're making the game for a perfect car lot (non DLC), these are the cars you should keep:
1. Berkley Kingfisher 9-15 106mph/225hp/2D
2. ISW 508 11-15 120mph/180hp/2D
3. Potomac Indian 7-15 106mph/170hp/2D
4. Shubert Beverly 8-15 103mph/190hp/2D
5. Shubert Pickup Hot Rod 15 110mph/209hp/2D
6. Smith Thunderbolt 11-15 121mph/233hp/2D
7. Walter Hot Rod 15 143mph/346hp/2D
8. Lassiter Series 69 2-10 107mph/192hp/4D
9. Smith Custom 200 9-15 118mph/228hp/4D
10. Walker Rocket 11-15 130mph/200hp/4D
This is with the belief that you can store the Hot Rods, I'm not sure if you can or not they spawn very very rarely in only two locations, I didn't waste the time to find them. If you can't then the Shubert Frigate and the Ascot Bailey are the next best two cars.