There are no cars in the community because everyone has bicycles.
Access to vehicles of any kind is controlled very strictly in the community. Since they have instituted Sameness, there are no hills and there is no weather, so there is no real need for a vehicle.
"And hills, too," he added. "They made conveyance of goods unwieldy. Trucks; buses. Slowed them down. So--" He waved his hand, as if a gesture had caused hills to disappear. "Sameness," he concluded. (Ch. 11)
Without hills or weather, there is no need for cars. They have some kind of transportation, but we do not know what it is. Most of the time everything is just referred to as a “vehicle.” Produce is referred to as being “trucked” from fields (Ch. 13).
As we know, everyone gets a bicycle at age nine. This is a big deal and a kind of rite of passage. Bicycles are symbolic in the community since there are no cars.
The bicycle, at Nine, would be the powerful emblem of moving gradually out into the community, away from the protective family unit. (Ch. 6)
Even adults only ride bicycles, except for a few exceptions. There are some cases where other vehicles are used. We know that they have cargo planes and jets, because a jet is mentioned in the beginning of the book as panicking everyone when its novice pilot accidentally flies over the community. There are other instances where actual vehicles are used to transport people to neighboring communities.
Consider Jonas and The Giver's first escape plan, which involves The Giver's access to a vehicle.
In the early morning, The Giver would order a vehicle and driver from the Speaker. He visited the other communities frequently, meeting with their Elders; his responsibilities extended over all the surrounding areas. So this would not be an unusual undertaking. (Ch. 20)
The Giver can ask for a car (notice that he also has to ask for a driver) because he is special. Normal citizens likely cannot.
By limiting who has access to vehicles to only those with special privileges, you limit the movements of community members. After all, there is only so far a person can get in a bicycle. This is why Jonas’s first plan of escape involved the “vehicle.” His actual, riskier escape was on his father’s bicycle.