Yes and no... Mostly no... Kinda depends...
Let's start with a stereotypical group of rural ranch/farm kids from central Montana. They've been in 4-H and Future Farmers of America for years, they've been members of the Boy Scouts and the Girl Scouts. Their parents annually bottle (can, store) fruits and vegetables grown in their own fairly large garden and they work with animals from rabbits and goats to cows and horses every day. They've been hunting with their parents since they were five years old. Could you drop 20-50 of these kids alone somewhere in a post-apocalyptic world and expect them to create a functioning society?
No guarantees, but probably yes.
Now let's consider a group of stereotypical city kids. They might be involved with the Boy and Girl Scouts, but the focus in that case is very, very different. They've spent a lot of time shooting hoop and playing video games. They don't raise chickens... in fact they can't (it's against city code), so the closest they've ever come to gathering food is buying it at a supermarket and with uncommon exception the closest they've come to preparing it is tearing away the cellophane and placing it in a pan on an electric stove. These kids don't go camping, they hang out at the mall. Could you drop 20-50 of these kids alone somewhere in a post-apocalyptic world and expect them to create a functioning society?
No guarantees, but probably not.
But whether or not they could, whether or not it's feasible, isn't a worldbuilding question. You're the author. If you want it to happen, it will.
I almost voted to close this question
Because it's too vague and too story-based. Is it feasible? Of course it is — depending on the ages, maturity, education, and skills of the children. Too little of all of the above and survival of a society becomes unlikely. Too much and, well, they'd be controlling Big Town like a shadow government.
All of which is your job as the author to determine. Because age, maturity, education, and skills (or their lack) are also what will contribute to the challenges they face and how they overcome them (if they do). But there is one thing I need to warn you about...
Children are not small adults
Teens, for example, take insane risks not because it's cool, they're capable, and adults are foolish for thinking they can't. They do that because very simplistically, their brains have developed to the point of better understanding the greater world around them but have not developed to the point of realizing that life is fleeting. Stories that present children as small adults cater to the children, meaning they're trying to sell books to kids, not to adults who know better than to believe that an arbitrary group of children are more than capable of establishing a functioning society and solving, not just adult problems, but those problems for the adults.
How many children would know what to do if they cut themselves? Broke a bone? Suffered internal injuries? The more dramatic the damage, the less likely a child can fix the problem. All those kids I just mentioned? Even those from rural Montana? Yeah, they depend on clinics full of adults who know what to do in emergencies.
In fact, most children depend on adults to know what to do in emergencies. The younger the child, the more this is true. We praise the rare child who keeps a cool head in an emergency — but in all but the rarest cases, what the child did was call 911 or another emergency services system to bring the adults to bear.
So, it depends...
Which is why I almost voted to close your question. Do you want help determining the skills necessary for such a society to exist? We can help with that!
Can one arbitrarily exist? That's storybuilding — a choice you, the author, must make, because outside of fiction there simply isn't an example of an independent society of children that functions at all.
Before you compare my answer to Nepene's (which is a good answer!) Consider the following: There were only six boys - that's not a society. They were close friends. They had strong faith, strong morals, and good discipline. They lived in a society that was much closer to today's rural societies than today's urban societies. The island they landed on had been occupied a century before, leaving them with a remarkable supply of food. And they were all older teens with no girls among them. This is really my point. Given the correct set of circumstances and skills, yes. Given an arbitrary set of skills and circumstances? No. Set that community next to an adult community... better hope the kids don't have something the adults want.