Create and populate the following file to add a dock shortcut to a website in Ubuntu using Ars Technica as an example. This will also enable the shortcut in the Gnome Shell search.
~/.local/share/applications/MyWebsite.desktop:
[Desktop Entry]
Comment=Ars Technica
Terminal=false
Name=Ars Technica
Exec=firefox https://arstechnica.com/
Type=Application
Icon=applications-internet
NoDisplay=false
To use Google Chrome instead, replace firefox with google-chrome if it is installed.
[Desktop Entry]
Comment=Ars Technica
Terminal=false
Name=Ars Technica
Exec=google-chrome https://arstechnica.com/
Type=Application
Icon=applications-internet
NoDisplay=false