A somewhat cold-blooded analysis of Vault 11:
First of all, in my estimation there was no reason for the inhabitants of Vault 11 to doubt the threat of its computer to exterminate the entire population of the Vault. The rest of this analysis will proceed from this assumption.
According to the Vault article, a vault could hold one thousand people, and the average life expectancy of a vault dweller was 92.3 years. This means that for a full vault, on average nearly eleven people would die each year, a rate that dwarfs the one sacrifice per year required by the computer of Vault 11. Indeed, if the inhabitants of Vault 11 chose to do so, they probably could have selected a terminally ill person each year--still morally unpleasant, but this would have kept the loss of years to a minimum, and not have even required an increase in the birth rate to offset as the person sacrificed would have been beyond breeding state. Therefore, we can conclude that the yearly sacrifice did not, in fact, even approach presenting a threat to the survival of Vault 11 as a whole.
However, on the social level, the sacrifices became not a burden, but a resource--a socially sanctioned method of murder. It was the heavy demand for this resource, and the attendant social, and eventually physical, violence that ultimately doomed Vault 11, rather than the yearly sacrifice itself. The yearly sacrifice would have cost only one or two terminally-ill months of life every year over the entire Vault; the fighting over the sacrifice all but wiped out the Vault and its future generations forever.
What would have happened realistically? I can't say for sure. I doubt a Vault would have wiped itself out entirely even under these conditions. Still, I would not be surprised if more years of life were lost fighting over the yearly sacrifice than from the sacrifices themselves.