If you are using the Vertical Menu with Sidebar or Popup or Overlay mode, you can open the menu by click on a menu item. First you need to create one item in the core Joomla Menu manager, or even you can use an existing one. As type you can set the System Links/External URL. Place a '#' into the Link parameter. Then you will need to define the menu item's Link CSS Style under the Link Type tab. Add the 'openmenu' class there.

Menu manager
Menu manager

Next you need to navigate to the Module Manager, select the Vertical Menu from the list you would like to open with the menu item, and click to edit. You need insert the following code under the Custom JS parameter:

$(".openmenu").click(function(e) {
 if(sm.popup) {
 } else {

After you pasted the code snippet save the module, check how it is working on your frontend.