I'm experiencing the same problem.
A few days ago it was still working but when I tried to download an app this morning I got the "403: Forbidden" message. I installed the new version of the extension (1.4.2) as recommended in this thread.
When I tried again I received a different error. I'm not sure if I remember it right but I think it just said something like "Can't download this app", no specific error message.
Reading on in different threads I found a post that said using the Google Services Framework ID was better than using the Android Device ID, so I typed in the GSF-ID instead of my Android Device ID and now I'm back to the "403: Forbidden" error. What's interesting, is that I cannot go back to using the Android Device ID because it only has 15 characters, so I get an error message (has to be exactly 16 characters) when trying to put it in the settings. I did a hard reset a couple of days ago but I was assuming the Android Device ID did not change with a hard reset...? Why did the extension not show the "16 characters" message when I first set up APK Downloader?
So if anyone has an idea on this issue, I would appreciate your help.