Amazon.com Inc. will begin allowing third-party sellers on its platform to resume shipping so-called nonessential items this week, a signal that the company is ramping up to meet broader consumer needs, according to people familiar with the matter.

Last month, Amazon made a decision to prioritize at its warehouses those items deemed essential during the coronavirus outbreak, such as cleaning products, health-care items and shelf-stable food. Amazon stopped accepting shipments of items from sellers that didn’t correspond...