I ordered some gifts—a hand salve, some tea, soap and a cookbook—from a Washington, D.C.-based boutique on March 19, a Thursday. My friend’s birthday wasn’t for another week and I figured the package could take a while. But it arrived on her New York doorstep two days later. I was stunned. The same cookbook on Amazon is currently delayed until late April.

Massive order volume, fueled by the global coronavirus crisis, is causing a logistics backlog at Amazon. Shoppers accustomed to receiving packages in a matter of days—the...