Is it possible to use the command line tool in PHPstorm to run Drush on a Vagrant box?
If so how do I go about this?
Command Line Tool plugin -- no and there is no current plans for that.
https://youtrack.jetbrains.com/issue/WI-23740 -- watch this ticket (star/vote/comment) to get notified on decision/progress.
But you can use built-in Remote Terminal (over SSH) for that: https://confluence.jetbrains.com/display/PhpStorm/Using+the+PhpStorm+built-in+SSH+terminal+and+remote+SSH+external+tools