There can be no generic, useful research on the social and educational impact growing up without a car has on children.
What makes generic research difficult is that there is no generic environment. The question about using a car is entirely different if asked in New York City or San Francisco, and different yet again in Los Angeles. Even within each of these cities it makes a difference where exactly you live (e.g. you don't drive in Manhattan). And that's just large cities in a single country.
The answer is, the generic question cannot be answered. It would need to be specific.
For one specific case: When and where I grew up, kids didn't even know if each other's parents owned a car, so it simply didn't matter.
PS: Children are teased if they are different from the "norm". If having a car is seen as the "norm" or not depends on the other kids. You can't avoid your kids being different from the "norm". Other kids will certainly find something else to tease your kids about (red hair, wearing glasses, being short, being tall, being bad at school, being good at school, having too many friends, etc). And your kids will do the same to others.
PPS: It's somewhat likely that society's take on cars will change in the next 20 years, in urban and suburban areas. There's a chance that extremely affordable self driving taxis, which will presumably be available by then, will make many people think twice about spending $15'000 every 5 years on a machine that spends most of it's lifetime sitting idle in an expensive parking space.