I’ve narrowed the issue down to the jQuery in rmp-menu.js (specifically the event.preventDefault() on line 189. When that is removed, the keyboard will reach the search input.
I’ve noticed another accessibility issue regarding the keyboard. Tabbing each parent automatically opens it and requires the user to tab through each link. This requires a user to tab multiple times to get to where they are going. A better way to navigate would be to have the tabs navigate to the parent menu and have an ENTER keydown expand it, then allow the tabs to move through when a user wants to.
Keep in mind that screen readers will allow a user to move through the links easily and gives extra help to blind users, but someone only using a keyboard, while still retaining their sight, will be continually forced to go through a series of links they may not be interested in and have no way to reach the search input.
Hope this helps
Sandra Clark
Shayna Productions
Apologies for the slow response, I have been traveling. The RM version is 4.33.