How to Emulate hardware menu button
Apps that could either customize the navigation bar or generate another way of navigation would get it done; however, I've yet to seen one emulating MENU key without requiring root permissions.

LMT Launcher is the app of this type I've been using - it hides a pie-shaped control tray into the edge(s) of your screen that you can activate by swiping from the edge, and contains not just regular navigation keys like BACK, HOME, MENU but also custom keypresses (e.g. POWER) and actions.