Taxonomy overview search term
Views with exposed filters sound like a great way to add a Search page to Vocabulary configuration tabs. 

You can add your Views page to Vocabulary pages through Views configuration. Since the List already exists as the default tab, simply configure your Views page Menu option as a normal Menu tab and set the page path to the [path-to-default-vocabulary-page]/search. 

The tab will appear as one of the tabs you see when you visit the Vocabulary configuration page.