How do you feel about installing the Silverlight plugin? I'm not sure why Amazon doesn't support the HTML5 video player in Firefox, but it should play videos in Silverlight if you install it.

https://www.microsoft.com/getsilverlight/get-started/install/default.aspx

If Silverlight is installed and on the Add-ons page is set to its default setting to "Ask to Activate" then you can check the site-specific permission for Amazon.

While viewing a page that should play on Amazon, check for the plugin notification icon in the address bar -- it typically looks like a small, dark gray Lego block. Click that to confirm Amazon has permission to use Silverlight.

If the block does not appear, you can use the Permissions panel of the Page Info dialog. You can call that up using any of these:

• right-click a blank area of the page and choose View Page Info > Permissions
• (menu bar) Tools menu > Page Info > Permissions
• click the padlock or "i" icon to the left of the site address, then the ">" icon, then More Information > Permissions

Near the top, you'll find the "Activate Plugins" section and you can set Amazon's permission to Allow for Silverlight.

(Note: there's no save button on that panel, your change should be saved immediately.)

Then reload the page (Ctrl+r) to see whether it worked.