I was able to read DAISY files from bookshare.org on Kindle Fire 7" tablets by sideloading the Go Read app.
My SO teaches special ed, and got Kindle Fire tablets for her students through Donors Choose, not realizing beforehand that Amazon's app store didn't have apps to read DAISY files. However, we were able to install Go Read on the tablets through sideloading them (it doesn't require rooting the tablets, but is more complicated than using the Amazon store). In retrospect, tablets using the Google Play store would have been easier (but more expensive).
I'm replying here even though the thread is old since this is one of the top hits when searching for DAISY files on Kindles.
Here are the steps we used to sideload Go Read:
First, you have to get the APK (like an exe or application file). I got the APK by installing Go Read from
the Google Play store to a non-Amazon tablet, then using "adb" (a developer tool) to get the APK. For convenience, here's
a download link for the Go Read APK (fetched 2017 Jan 1). You'll need to unzip this file after you download it.
Then, sideload the APK. Summarizing
these instructions: (1) install ES File Explorer from the Amazon app store, (2) use
the Android File Transfer app to copy the APK into "Downloads", and then (3) tap to open the APK file from within the ES File Explorer app. (You'll need to turn on "Apps from Unknown Sources", but the tablet will prompt you to do that when you go to install the APK. Note that the instructions linked above suggest installing Google Play Services and other APKs, but you don't need to do that for Go Read to work, they're just necessary to support Google apps like GMail.)
Edit: bookshare.org has
a Help Center article which basically says DAISY doesn't work on Kindle, but
a thread on their forum both mentions sideloading and says they're trying to get Amazon to allow Go Read in their app store.