There's an experimental way to do it:
Enable the "Custom user sites for browser sidebar" flag at the URL opera://flags/#sidebar-site-panel, restart Opera, click the 3 dots at the bottom of the sidebar, scroll down to "Custom Site Panels" and add the URL for the web app.
There are a few catches though:
You can't change the icon for a site. Opera just uses site.com/favicon.ico.
You can't move the icons to where you want on the sidebar. They'll be at the bottom away from everything else and in the order you created them. You can't change the order by dragging them around.
You're limited to 5.
Site panels are restricted to a single domain. If you click on a link on the site panel's page that points to a different domain, Opera will open up that link in a new tab instead of navigating to the link in the panel. And, if the panel page tries to redirect to a different domain, you'll get an error. For the latter, you'll get this for sites that redirect you across domains to log you in. In this case, log in via a tab first and then reload the panel page.
