As this couple says, being without a car isn't for everywhere and anyone, but in some places it's perfectly fine to do even if you have kids. Also, too, before relatively affluent white people started arguing about it plenty of poor people raised kids without cars because, you know, cars are expensive.
I don't have kids, but most of the "reasons" I always heard for why having them without car would be unpossible seemed silly. Its appeal depends on a lot of factors not always within your control, such as your work/commute situation. But with a bit of luck and some choices it's fine. And one thing that people seem to forget about kids is that they grow up fast, in the sense that what might be a pressing need at 6 months isn't so much of one at 2 years, and probably won't be a big deal at all by 5.
If most of your dailyish and regular baby needs (food, pharmacy, daycare if using, doctor) are within a 15 minutes walk, being without a car is fine. Cars are useful things, of course, but you save a lot of money by not having one and even if only half of that is whittled away on extra taxis/car share/weekend car rentals, then you come out ahead. Bulk buying/delivery (either from supermarket services or amazon) can ship diapers and similar to you (one of the big objections people make, weirdly - how will you carry home all of those diapers?? that one is easy).
None of that means everyone should go without cars. Make your own choices! I'm really not big on telling people how to live their lives, and we rarely understand the constraints other people face. But in discussions about urban hellhole living one often hears "wait until you have kids!" - both with respect to living here at all and living here without a car. Even the people who run the city assume all the "newcomers" will leave as soon as they have them, which isn't exactly ideal for public policy choices. Both are clearly possible. Whether they're desirable for you, personally, is another question.
If you live in a neighborhood like mine (a big if) and having a transit accessible job (another big if), it really isn't a big deal to be without a car, with or without kids.