I recently went through this myself.
By default, Chrome uses Google's version of flash through Pepper API (PPAPI). Amazon doesn't work with that version. Since it doesn't work in Firefox, you must not have Adobe's version of flash installed. To get Adobe's, install "flashplugin-installer". The name is misleading, it will actually install the plugin itself (Due to licensing issues, the binary has to be downloaded directly from Adobe, so you're technically installing a script that automatically downloads and installs flash).
After that, Firefox should work fine. To get Chrome to work, you'll also have to disable the PPAPI version of flash:
- enter "about:plugins" into the address bar of Chrome
- click Details in the upper right corner to have it show all the files
- Scroll down to flash and disable the PPAPI version