Absolutely, it is possible to import your website’s application to the Softaculous tool as long as it is one that Softaculous supports.
All you need to do is log into your Softaculous from your cPanel, find your application type there (the one your site was built on), and hit the import button. Say, you have a WordPress site:
If your site is in a subfolder and accessed by yourdomain.com/wordpress/, for example, you need to specify the directory. The directory in this case would be /wordpress/. If the site is in your main public_html folder and accessed by yourdomain.com, you need to leave the field empty.