This plugin has its own fluid system of required user capability definition.
With Admin role selected at Plugin access level it is enough to have ‘manage_options’ capability to get full access to this plugin.
With Editor selected at ‘Record Edit Access Level’ drop-down menu it is enough ‘edit_others_posts’ to get access to the list/add participants menus (screenshot attached).
For ‘Contributor’ selected it will be ‘edit_posts’.

Version 4.18.b2 updated this way:
Admin menu items with empty user capability are available in “Admin menu access” add-on now. “Participants Database” plugin defines its menu this way.