Package-delivery companies are bracing for their busiest week of the holiday season, and one with less room for error.

FedEx Corp., United Parcel Service Inc. and the U.S. Postal Service expect to process a record number of packages this peak season, many of them e-commerce orders, and the week before Christmas is usually the heaviest period. The Postal Service, which makes more deliveries to homes than its two commercial rivals, already plans to exceed its initial forecast of nearly 850 million packages between Thanksgiving...