New York terror suspect Ahmad Rahami assembled much of his bomb-making material in plain view—ordering components on eBay, having them delivered to a New Jersey business where he worked, and even testing some of the material in his family’s backyard, according to authorities.

Based on the evidence collected to date, a top priority for investigators is understanding whether Mr. Rahami’s immediate relatives knew about his alleged plotting, law-enforcement officials said.

“It’s...