Hi there

You can add text and html to widgets but not really php. I think the easiest way to get the effect you’re looking for would be to get a modal plugin (http://wordpress.org/plugins/easy-modal/) and then put the shortcode to trigger the modal in the feature content widget, that way when someone clicks on the widget content a modal pops up over the screen with the content/booking form in (maybe something like content form 7).

If you’d rather have it so additional content popped out from the side of the page, then really I think you’d need to contact theme customizers on http://www.themecustomizers.com/ to see if they’d be able to help you come up with a custom solution.