Transparency and communication, including real-time delivery tracking and notifications, are also key factors for making millennials happy with the shipping and delivery experience.

Rod Sides, a vice chairman and U.S. retail and distribution leader for Deloitte, said millennials are setting the definition of what fast delivery means, in an annual survey the consulting firm conducts.

"It always comes back to the millennial mindset that as a group they want what they want, and they want it immediately," said Sides. "They're willing to pay more (for expedited shipping), and see a value in getting orders quicker, understanding it's a tradeoff. Also millennials are not spending as much time looking for a deal as they are looking for instant gratification."

The Deloitte study, which surveyed 5,000 consumers in September, found that millennials were willing to pay an average of $5.50 to get their order the same day during the holiday season, compared to an average of $3.80 for non-millennials. For next-day delivery, they were willing to pay an average of $3.90 per order, compared to $2.60 for non-millennials.

Millennials also have a different idea about holiday season cutoffs, according to the Deloitte study: 69 percent of them said they believed they could order something after Dec. 17 and still get free shipping, compared to 61 percent for all other consumers.

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