It's possible to setup side tab on your website. Click on it will lead user to the popup where it will able to leave feedback or search for an existing ones.
Below on screenshots, website screenshot with tab widget installed we point it with red arrow. Widget popup and widget tab settings.
Replace your contact form with our inline widget, that allow you to manage all feedback from one place.
More details about inline widget.
Provide forum_id param to open specified forum
You can get forum_id via Setup->Community Forum->General->ID
or Setup->Helpdesk->General->ID if you want to open private message form
You can get widget embedding code via following path in the settings:
Setup -> Widgets