It's VERY implausible that even if there were ways to repopulate LL that the children would be even remotely near to what we know children to be today.
They grew up with children barely older then themselves to be there parents thus the maturity level that you see in LL would be virtually non-existant.
Not to mention that with virtually no concept of god an afterlife or what death means the LL kids would most likely be a band of savage half-pint canibals.
I mean have you ever really thought about what kind of social order would evolve if you threw 50 9 year olds out on their own? Utter chaos my friends....
Not necessarily... Societies are often built out of the good and welfare of the whole. The people willingly submit to some governance out of the understanding that it allows for their to be some level of equality and control. If it was a complete free-for-all, then it couldn't really be regarded as society or having any social order. It would be anarchy. Even with 9-year olds, there would be some order. If nothing else, the strong would bully everyone else into submission and run things. Little Lamplight having a more democratic society might be less plausible, but not impossible. Due to the threat of bullies doing whatever they want to the other kids and eventually destroying the society, everyone else could have banded together. Then, just as in most societies, the older children (before leaving) could pass down the knowledge to the others. The rules and societal structure could thus be preserved and carried on throughout the generations of children arriving. This doesn't mean they could maturely handle all situations, but adaptations could be made to suit the needs of the society. In other words, out of need comes innovation. Children are capable of acting maturely and responsibly, it just shouldn't be expected to be all the time.
There really isn't that big of an issue with repopulation. Say somebody in Nova's line of work (from Megaton) has a child. Do you think she would keep it, or send it to Little Lamplight? Same goes for others in the Fallout universe. It is quite probable that settlements close to the capital wasteland would send their children there if there was any doubt about their ability to raise and protect the children. There are very few secure locations in the wasteland. Children could easily become a distraction that reduces survivability. Even if those getting kicked out of Little Lamplight hate the place because of that, there is still a good chance they would send any of their children there. After all, the settlement they move to is one of the least secure in the capital wasteland.
The whole teenage pregnancy thing probably wouldn't work for long term repopulation though. Without adequate hospitals and facilities, the mortality rate for those giving birth (especially at a very young age) would probably jump sky high. The mortality rate for advlts would also jump up in this regard; however, that would likely lead to the children of any passing away while giving birth being sent to Little Lamplight. It really depends on the community and the society, as well as surviving family members.
I think one of the best examples I can think of where children would likely be sent away when possible is Rivet City. Even though they are a decently secure society, they don't hold children in high regard. There are also very few children in the society. If an advlt passed on, leaving behind children with no one to watch, they would likely quickly be sent off to Little Lamplight. Just speculation on my part, but Rivet City seems to consider children a distraction and an annoyance.
There is also merit to some of the other ideas that have been thrown around. This was just my take on it.