Just tried out an app from 1mobile called Shake - Lock which seems to work on the Fire. I found it works more consistently if you shake the top end of the Fire up and down. This locks and unlocks the rotation. In the settings you can set an icon to pop up and show when it successfully changes from lock to unlock or vice versa.
It's a little tricky to adjust the sensitivity.
Anyway, it's a bit easier than having to go up to the Kindle top settings and change the lock/unlock.
Edit: Sorry, this is a widget for Go Launcher EX so not usable on the stock KF.