The Zendesk help center provides end users with a default HTML form for submitting tickets:
Customizing the help center ticket form
Embedding a ticket form in your website
The embeddable Web Widget (Classic) includes a customizable HTML form that lets end users submit tickets. You can embed Web Widget (Classic) in any page of your website. Clicking the widget gives the user the option of using a ticket form:
Building a ticket form
For maximum flexibility, you can build your own HTML ticket form, host it on your own server, and use the Zendesk API to submit the form data to your Zendesk Support account.