Adobe stopped supporting Flash for Android devices in August 2012. This means Amazon cannot provide Flash for the 2nd Generation Fire and the Fire HD models. (All Kindle Fires are Android devices)
There is a work around though. To view Flash videos, you will need to install an earlier version of the Dolphin Browser and install an earlier version of Flash.
Note: To watch Flash videos, you'll need to watch them via the Dolphin Browser after everything is installed properly, and NOT via the Web section (Silk Browser) on your Fire.
Installing Adobe Flash Player On Newer Kindle Fires
(2nd Gen Fire. and HD Models)
If you have a newer Kindle Fire model, Flash has to be installed manually, along with a Flash compatible web browser.
To install Flash: (You have to have a Flash compatible browser in order to install Flash. You can't install it under Silk.)
Do the following on the Fire/Fire HD itself:
Swipe down from the top > More > Device > Allow Installation of Applications.
Install ES File Explorer: ES File Explorer
Download and install the Dolphin HD and Flash Player apk files from these links:
Once both files are downloaded to your Fire, open ES File Explorer and look in the "Downloads" folder. You should see both files there. Install Dolphin HD *first* by tapping on the icon. Then install Flash.