Stigler reports that he received a letter from Tjallling Koopmans asking whether he had in fact advocated the use of the price system to evacuate NYC in case of a bombing during WWII. Stigler responded that he had never even thought about the problem before, but upon reflecting on the problem that (1) upon the first bombing of NYC any system of evacuation would be chaotic and inefficient, but that (2) if the bombings were repeated, that indeed he would argue that the price system would be the most efficient way to handle the problem.
It depends on the counterfactual. It is already the wealthy who have the resources to leave afflicted areas, or who had the resources to send their children to the countryside, in the case of bombed London during World War II. You could pay a relative to take the kids in, rather than having to rely on the charitability of your relatives. So very often we already are using the price system, and in a fairly orderly manner.
If you are evacuating a city suddenly, along a constrained road or path, ideally (at least by economic standards, which may or may not be your final moral theory) you wish to favor the people who are young, productive to others, and people who value their own lives highly and are risk-averse. A market auction tends to favor the wealthy and in this context many of the first leavers in line will be inefficiently old, again with the moral caveat noted above. The wealthy spend money on the basis of "if I die, my wealth is worthless or worth less because my bequest motive is less than full," whereas from a social point of view the wealth survives the death of the wealthy person.
If institutions will enforce the traditional "women and children first," with a minimum of corruption, that solution may be preferable to the auction. Men are on average more productive than women in the labor force, but the number of replacement children in the longer run is more closely tied to the number of women than to the number of men. So, indirectly, favoring women favors men too.
The private sector often chooses the rule of "women and children first," at least when the disaster is explicitly seen as such. This rule was heeded in the case of the Titanic but not the Lusitania, arguably because the latter ship sunk more quickly and with more panic.
In many other settings, especially where dying is non-immediate or stochastic, the market chooses the auction method. Think of the market for pharmaceuticals. In the absence of government subsidy, you have to buy the drugs and there is not always price discrimination in favor of women and children. Also consider allocation procedures for kidneys, hospital rules for triage, and the sale and resale of fresh water during cholera epidemics, among other scenarios. What's striking is how many different allocation procedures markets use, depending on context.