FEMA was formed 1979, but before that, was there a certain agency that handled natural disasters? Did the federal government just send monetary aid to the state governments and let them handle it?
Relief for the 1906 San Francisco earthquake was provided by the US Army, whose support mainly involved distributing food/supplies to survivors, protecting businesses from looters, aiding the fire department in dynamiting buildings to create fire lanes, and establishing temporary housing camps for those whose homes were destroyed.
About 4000 troops saw duty in the disaster over a period of about a month and a half, after which the Army withdrew from the city and civic authorities in San Francisco assumed responsibility for the relief efforts.
You can read Maj. Gen. Greeley's report on the relief effort here.