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 Latvia, Lithuania, Estonia, Kazakhstan, Georgia, Uzbekistan, Moldova, Armenia, Azerbaijan, Tajikistan, Turkmenistan, Kyrgyzstan, since Amazon 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