CASAMICCIOLA TERME, Italy—Angelo Croce was playing in a table-soccer tournament in this village on the island of Ischia when the ground began to shake violently. He ran home in a panic.

He had reason to worry. The two-story house, built without permits, lacked earthquake reinforcements, even though Casamicciola sits on a dormant volcano in a zone susceptible to mudslides and quakes. As debris fell, Mr. Croce passed a woman fatally injured by a collapsed church facade. His family was unharmed, but still can’t return to their...