I know that some Fire tablets are being used with BridgePal by some of the welsh clubs, but I believe they have replaced the amazon firmware with a more standard implementation of android. For instructions on how to do this see
this link, which contains instructions for the Fire 1 and Fire 2 tablets.
You don't actually need google play store access to install BridgePal on a standard Fire tablet because you can download it directly from the BridgePal website (see the link in the section on Scoring Unit Setup). You would need to enable installation from 3rd party sources in the Fire tablet settings in order to do this. However, I think this approach is less satisfactory because the Fire tablet is heavily customised (e.g. adverts on the lock screen). Also, it isn't possible to make BridgePal the home screen on a standard Fire tablet (this is a new feature in BridgePal Release 1.5.4)