This is easy. They do home births, perhaps having one or two midwifes/obstetrics-nurses/obstetricians in their group.
They can opt in or opt out of birth certificates if they so choose (whatever is convenient for the storyline). If they opt out, special precautions are taken to avoid the sorts of issues that creep up when you have young children without that documentation (long distance travel, primarily, if you have nosy cops asking to prove that the baby's yours when they pull you over for a broken tail light).
Social security numbers are likewise not an issue (even if you have the baby at the hospital, it's a piece of paperwork you have to fill out to get one).
The children will be homeschooled. If someone notices this, it's not completely out of the ordinary nowdays... so they can hide among all the rest. It won't raise many eyebrows.
Extracurriculars are limited to those where group photos will not be taken (or maybe eliminated entirely).
They'd live in flyover country. Nebraska or South Dakota or Idaho. The kids only have friends from parents who do not use Facebook and don't want their own kids' photos posted to social media (I know some of that sort of people myself).
This makes it easy for the children to learn to use firearms without going to a public range (where photos are occasionally taken and posted). Other espionage-adjacent skills are more difficult to pursue in a social-media-absent way. Flyover country's not great for most martial arts, as I've discovered for myself.
The real trouble comes along when you're trying to set up multiple fake identities for each child. Since 9-11 this has become very difficult (though not impossible) in the United States. It's possible through some states' vital records departments to register birth certificates online. These aren't always rigorously verified either. Registering them for a social security card is difficult and time-consuming too, and some SSA folks are nosy enough to want to see the baby when you get the number. Those two documents are the basics, but more is needed for a false identity that is unremarkable. Likely, in this day and age, those children's false identities will need a social media presence.
The best bet is to never use their photographs. Instead, one would use GNN-generated "people who have never existed" for this purpose. The technology's not quite ready off the shelf... it has trouble generating the same fake person twice, for instance. Or that the ones I've found can't generate children of arbitrary ages. Though one could take a single good image from that, and deepfake it into other images for reuse. Since children's faces are assumed to change quite a bit growing into adulthood, you need only match hair color, eye color, and skin tone... the pictures won't look like the actual kids, so pose little danger of exposing them early.
It might be a full-time job just creating these images and false personal histories and posting those to Facebook/Instagram/whatever. Might be more than that, so I hope your secret society has plenty of resources. Image files have to be created with fake exif data, uploaded hours within when they were supposedly taken. Background imagery has to be sanity-checked (you don't want to claim it was taken in Portland Oregon at that intersection without knowing for certain no local news events occurred that would make that location absurd... e.g. a large housefire had 18 firetrucks parked right there for 6 hours).
Crap, if anyone wants to downvote this answer go for it. I only just realized that this requires that the group start operations in the 1990s... some of this knowledge did not exist then. But I'll post it anyway to show that it could be founded much later and still be threadbare plausible.