I was having the exact same problem with my lumia 640 the only solution i could find was to upgrade to Windows 10 which has the option to allow apps from unknown sources so you dont have to rely on the microsoft store
This is how to upgrade to windows without using the upgrade advisor from the store:
first download the over the cable updater tool from microsoft using the link below THIS MUST BE DONE ON LAPTOP/PC NOT YOUR PHONE
https://www.microsoft.com/en-us/download/details.aspx?id=56726
follow the instructions in the OTC Updater ReadMe Final file and this will upgrade your phone to windows 10
Basically all you have to do is turn your phone to airplane mode and connect it to your laptop/pc via USB and run the program (it takes quite a while and the phone will restart a few times)
once you have window 10 installed here's how to get apps from other sources:
From the start menu go to settings Select “Update and security,” and then select “For developers.” Activate the “Sideload apps” (this is the same as if you select unknown sources on an android phone)
As support is ending for windows phones next month i suggest checking for and downloading software updates as often as possible between now and 10th December
Hope this helps, because it looks like the stores not coming back