The Zendesk for Wordpress plugin includes a comment form that you can add to the Wordpress administrator dashboard so that your site users can easily submit support requests (see Setting up and using the Zendesk for Wordpress plugin). By default, only your registered Zendesk users can submit requests. You can however allow your Wordpress site users who do not also have a Zendesk account (considered anonymous users) to submit support requests.
There are several easy steps to set this up:
Enable anonymous ticket submissions in the Zendesk for Wordpress plugin.
Choose an existing Zendesk administrator or agent who is also a user in your Wordpress site to act as the delegator for anonymous requests.
Have the assigned agent sign in to your Zendesk via the Zendesk ticket widget to authenticate their account.
This delegator role is simply a means to access your Zendesk and submit the request on behalf of the anonymous user. Depending on how your Zendesk is set up (open or registered users only), the requester may also be required to register with your Zendesk and create a login so that they can follow up with the ticket.
Note: Anonymous requesters are never aware that their request was submitted via this delegator; it's all done under the hood via the Zendesk API.
When you're setting the visibility permissions for your Wordpress site users (see Setting the dashboard widget visibility by user type), you should allow all agent users to see the ticket widget on their Wordpress dashboard so that they can sign in to your Zendesk to be authenticated via the widget, which is required to set up this delegation.
The delegation agent needs to login here before they can be assigned the delegator role: