Over the course of Eddsworld, Edd has owned many different cars, his most recent car being a red one similar to Tord's. However, he does tend to walk most places nowadays and the car isn't featured as much. In Edd Again, he takes the bus to town.
The Eddsworld mobile
The Eddsworld mobile first appeared in Zombeh Attack. Edd and the gang are out driving when the accidentally drive into a man fleeing from a zombeh attack. The car flies into a graveyard, lands on its back and breaks down. Edd and Tom quickly repair it and use it to flee the graveyard at the end of the episode.
It reappears in Zombeh Attack 2, this time without the writing on the side. Tord hijacks the car to force Edd to let him inside his house. Tord then accidentally slams the car door on his leg while getting inside, causing it to be sliced clean off. The car is used to kill Kim and Katya but ends up crashing into a wall in an attempt to hit Matt. This causes the car to break down so Edd and Tord abandon the Eddsworld mobile and go on foot to the necronomicon. After that, it is never seen/mentioned again.
The Blue Car
A blue car appears in The Dudette Next Door parked outside Edd's house for most of the episode. It doesn't appear again.
Casino Night Car
This car appeared in Casino Night, Edd and Matt use it to get to a casino, then afterwards they end up getting drunk when they lose all their winnings. Matt attempts to drive the car home but, because he is drunk, he ends up crashing it into a lamp post.
The red car
Edd's red car first appeared in Zanta Claws II. This car has a special Z Gear that enables it to fly. Edd uses this to get to the North Pole by car, however, Matt ruins the moment, causing the car to lose its powers and drop from the sky. Luckily, it manages to get a lift from several airplanes to get to the north pole. Edd then leaves it on top of a stack of planes while he deals with Zanta Claws.
Matt uses the car to look at his reflection in the mirrors, he sometimes drives the car while doing this because it he thinks his reflection looks cooler when the background is moving. In the comic 'Accident', Matt crashes the car through the wall of the house resulting in the story arc of Tom being put in a full body cast. At the end of the arc Matt crashes the car into another wall, this time destroying the car.
In part two of Hammer & Fail, Tom was thrown into the car, resulting in it being destroyed. Tord himself owns a red car very similar to Edd's red car which was shown only twice. Once being when he left the gang in 25ft Under the Seat, and the other when Matt tries to remember Tord in the first part of The End.
The same car appears in Fun Dead during the intro.
This car's license plate is 130985 SMB.
CDT-01 is a tank that has appeared in MovieMakers, Moving Targets, The Snogre and The End. It is a bit skeptical that Edd owns it but we have seen him steal it twice from 2 different military bases. It is also questionable if they were both the same tank, as more than one tank is seen when Edd steals it from the secret military base.
Edd's go-kart appears in a skit in Random Bits where he drives around a random woman in circles with it to the tune of "Jerk It Out". He always wears 3D glasses while riding it.
Three of these go-karts appear in WTFuture as part of Operation MegaEscape4, Edd, Matt and Tom wore 3D glasses, as well as fairground music which played in the background. The group used them to escape Future Edd, who was trying to break into Edd's house.
One of those cars appeared in the car park of a supply store in Hammer & Fail (Part 1). Presumably, this is what Matt used to drive himself to the store in the first place.
Similar bumper cars also make a short appearance in 'Fun Dead', In which Edd, Tom and Matt use them to escape the zombehs after the ferris wheel breaks down.