Is it possible to install Google Chrome on a Kindle Fire HD? I searched the Amazon store and didn't see it there.
Yes. According to the guide here on sideloading apps onto your Kindle Fire HD it should work fine. You will need to allow the installation of apps from unknown sources which can be done in the Device menu. Note that for certain apps, including Chrome, that need a Google account you'll need to download and install the Google Framework and to reboot your device.
I'll provide a quick summary of the link that explains side loading:
The first thing you need to do is install a file manager app. I personally use File Expert, but you can use whatever works for you.
Next you need to get the app file. You can find it by searching on Google: (app name) apk. So in this case you would search 'chrome apk', there's plenty of links from that.
Once you have those, navigate to the Downloads folder with the file manager, and tap the .apk file. Click Install, and you're set!
You can find the app in apps>device, it will not be in Cloud.