Hi, vsync.
This will be because in the theme’s style.css stylesheet the style for .widget-area #s
is set to be 95% of the container in which it is enclosed.
You can add a similar style to your Custom CSS box in Appearance > Theme Options > Custom CSS to target the search box you are using in the Visual Composer widgetised sidebar.
If you use the Internet Explorer, Chrome or Safari browsers you can use their built-in developer tools to point at and “inspect” elements on a web page to find out what styles they use. You can even tweak the styles live to experiment with any changes you might fancy. If you use Firefox then there is an add-on named Firebug you can install which gives that browser the same functionality as the developer tools built into the other browsers.
If you need to learn how to use CSS then I can recommend this site: http://www.w3schools.com/css/
If you need me to look up the necessary CSS for you then I will need a link to a live web page that dispalys the issue.