There is actually a nice general method:
google-chrome --app=https://web.whatsapp.com/
and this works for any website. So for example
google-chrome --app=https://pt2.allsup.co/linux/xdg/OpenWhatsappWeb
will open this page as an app. Note that such apps do not get a URL bar.
Then you can (thanks to gist linked below) use wmctrl to activate a Whatsapp window
and only fail over to launching a new one if there isn't one:
wmctrl -xa web.whatsapp.com || /opt/google/chrome/google-chrome --app=https://web.whatsapp.com/
# or
bash -c "wmctrl -xa web.whatsapp.com || /opt/google/chrome/google-chrome --app=https://web.whatsapp.com/"
Linux .desktop file
From this gist