The days of saying "the Amazon Appstore
requires sideloading and thus can't be used on AT&T phones" may be ending. AT&T is reportedly beginning to push out updates to some older devices to enable sideloading functionality on them.
Sideloading is a term that means the ability to install builds from non-Android Market sources. That could be another store like the Amazon Appstore, or it could be an APK file on an SD card. To enable it, you need to go into the device settings, and allow "Unknown Sources" to install.
, in its misguided wisdom, saw fit to remove the setting. Once the Amazon Appstore launched, it seemed inevitable that AT&T would cave.
The updates that are coming cover the HTC Aria, the Samsung Captivate, and the HTC Inspire 4G. The recently released Samsung Infuse 4G came with sideloading enabled. Based on AT&T's informational page
on the matter, at the very least the Motorola Atrix 4G will get the fix, as well.
The updates are coming OTA, which means it's hard to predict when they will arrive, but they are going out now.