If you have a rooted Android phone with the latest version of Flash installed, you can try this:
1. Make sure you have Dropbox installed and set up on both your Kindle Fire and your Android phone.
2. Using Root Explorer (or other similar program), navigate to /data/app. Find the Flash .apk (com.adobe.flashplayer-1.apk), long press it to bring up the contextual menu, and press Copy.
3. Still in Root Explorer (or whatever), navigate to /sdcard and then Paste the item there.
4. Open Dropbox on your phone, press the Uploads tab, press the Other files button and then select the file to upload. Make sure the upload location, represented by the wide button at the bottom of the screen, is where you would like the file to go.
5. After the file is uploaded, launch Dropbox on the Kindle Fire. Navigate to the file and press it to begin the installation. You will see a warning message regarding installation of an already-installed program...all is okay.
6. You should see a message indicating installation was successful.
I highly recommend Dropbox for sideloading...so much easier than hooking the Fire and your phone up to your computer and moving files around in that manner. Good luck!
NOTE: In order to install Dropbox on your Kindle Fire, you WILL have to make some physical connections, first between your phone and the computer, then between your computer and your Kindle Fire. But once that's done, you will virtually never have to make those connections again.