Dropbox is available in the amazon store, but not for my 2nd gen Fire, so maybe not for the HD either. If you have another android device you can get it that way and sideload, I'm sure, with the file es app. Or, you can get it from 1mobile.com or getjar or something similar.
I just read, you can just browse to https://www.dropbox.com/android
and download the apk, then install with es file explorer.