Comment 2 for bug 1688627

I have a minimal working implementation.

Try It
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sudo add-apt-repository ppa:jbicha/epiphany
sudo apt update
sudo apt install webapp-amazon
Log out then log back in (to run the required autostart)

Brief details
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This is using Epiphany's webapp mode. The epiphany packaging is directly inspired by Fedora which has shipped epiphany like this for a while. Firefox is still Fedora's default browser and although this includes /usr/bin/epiphany, you'll want to install 'epiphany-browser' if you want the .desktop file. (Epiphany calls itself "Web" these days as part of GNOME's minimal naming strategy.)

epiphany uses the webkit2gtk engine which is already in main and is well-maintained by Ubuntu Security for 16.04+. epiphany has no other universe dependencies and is a relatively small package (less than 2 MB download, about 8.5 MB installed including all translations).

How to completely remove it
---------------------------
sudo apt purge webapp-amazon
rm ~/.local/share/applications/ubuntu-amazon-default.desktop
rm -r ~/.config/epiphany/app-ubuntu-amazon-default

(and of course run ppa-purge as long as this is still only in a PPA)

Other web apps
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If we want to allow users to easily install other webapps that aren't part of the default install, it's a bit simpler. We just need to provide the appropriate appstream metadata to be able to use GNOME Software's Epiphany plugin.

I prepared one sample app for testing, but it is easy to add more components to the metadata file.

sudo apt install webapp-data
pkill gnome-software
gnome-software

Search for BBC News and install the app. The install is almost instant and it's only installed for the current user.