Those shipping companies almost always have some percentage of empty space on every truck, train and plane. Giving it to Amazon at even a tiny profit makes huge business sense.
Amazon also has its own shipping and air cargo operations, and local contractors that distribute parcels. It's not UPS or Fedex that's doing it any more, at least not the majority of it.
Not with the volume that Amazon has. Entire courier companies have been launched to deal with Amazon volumes. There's no way that the volume that Amazon has (and, say the 200 trucks a day at the proposed Island distribution centre) would have fit in the "empty space" of the existing infrastructure.