The original Fire could, I don't see why the HD couldn't. Here's a response from Amazon themselves on their forums last year:
You can charge the Kindle Fire while the device is off. The indicator light, however, should be giving you either an orange light if your Kindle Fire needs to charge, or a green light indicating that the Fire is fully charged. If you are not seeing the indicator light come on while plugged in, even if the device is off, please follow the steps below:
1. Make sure you are using the Kindle Fire charger that came with your device.
2. Un-plug and plug back in the Kindle Fire.
3. Restart the Kindle Fire (To reset your Kindle Fire, press and hold the power button on the bottom of the Kindle for a full 20 seconds.)