If you read the original post, you'll see that the user clearly said she had already tried setting her Kindle to "unlock", but that didn't do the trick. I'm sitting here with a Kindle Fire that's showing exactly the same symptom--in other modes it will automatically switch from portrait to landscape, but when using the web browser, it will not. And the settings have been reset to "unlock" multiple times. So the advice (basically, read the manual) was not helpful at all.
HT-Guy - This is the only thread I've ever seen with this problem. If I were you I'd contact customer support if you have to reboot to unlock your orientation.
TO CONTACT KINDLE CUSTOMER SERVICE: Click on the big orange 'CONTACT US' button to the right. You may have to sign in. I recommend you use the PHONE or CHAT options. Do NOT use e-Mail. CALL ME under the PHONE tab is great if available. They will call you within minutes.