You can use parcel/package forwarding services for forwarding your purchases from US online stores. Such companies provide you a virtual USA address. Some of them allow you to consolidate your packages, so you can save on delivery a lot. You will have the least problems with Amazon when purchasing from Saudi Arabia, Egypt, India, Pakistan, China, since it accepts international debit and credit cards.

See also:
- Major freight forwarders for shipping packages/parcels from Amazon & other US stores
- Online shops shipping and not shipping to Shipito warehouse in Torrance, CA