WASHINGTON—Less than a month into his new job, White House press secretary Sean Spicer needed to keep his food and drink cold. He wanted a mini-fridge.

He dispatched a top aide to a nearby executive office building where junior research employees are crammed into a room, surviving on Lean Cuisine frozen lunches. Mr. Spicer wants your icebox, the aide said, according to people familiar with the incident. They refused to give it up.

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