They don't need planes for the vast majority of their deliveries. Amazon has distribution centers in 21 states, within 20 minutes of something like 30% of the population, and within plausible single-day delivery range of probably the majority of the population. So for products that are frequently ordered (and thus are stocked at every depot), they can deliver to almost everyone on the same day by truck, or rent a few small warehouses in a few places and deliver the next day to probably 80–90% of the U.S. population.
Sure, they'll still rely on UPS/FedEx/OnTrac for deliveries that involve distribution centers in other parts of the country, or for deliveries out into the boonies, but that makes up only a small percentage of their deliveries. And for the ones that don't require all of that, there's no real advantage to using a third-party trucking service over doing it in-house.