Not exactly but close. I admit that private efforts have played a role in disaster relief. Calling people acting on their own volition outside government the market is incorrect as this is not a market driven reaction, it's people want to help. They are not doing it to get a reward or as a reaction to a market incentive, so this has nothing to do with the market. But sure, that happens.
Is it enough and would it be enough without the organized initial response from agencies whose job is to do that right away is another issue, but there are certainly people that get off their assess and volunteer to help and there are even more people that stay home but donate which is also help.