Relief organizations for floods, fires, etc., are a very modern development. I was just thinking of how they might exist in your typical fantasy setting.
The most obvious choice is the Temple of the God of Healing, whoever that might be. Helping the poor and needy is their divine mission. In fact, doing so will greatly improve the believers chances of going to Heaven.
A merchant organization could provide these services, for a fee. They might come in after a disaster and help everyone out, feeding people and even rebuilding lost homes. Then they give the local lord the bill. Of course, their fees might end up bankrupting the noble, but them's the breaks.
Maybe a wizard shows up, and he helps dispose of all the bodies. Maybe those bodies become experiments in his lab, or get reanimated as zombies. Best not to ask.
The local druids might help out, in an attempt to bring nature back into balance. Or maybe they are the ones who called down the destruction, to smite the city people and their loggers and miners.