Does Zendesk offer an embeddable contact form tool?
Our Web Widget will allow you to embed a simple contact form on your website, however, the customization options for the Web Widget are limited. See Using the Web Widget to embed customer service in your website.
If you want to embed a more complex form on your website, you should be able to create an HTML form and use our API to submit the request, see:
Also, the following apps can assist you in creating ticket forms: