Does Zendesk offer an embeddable contact form?
Our Web Widget allows you to embed a simple contact form on your website, where users can search for knowledge base articles, live chat with agents, submit a request, and request a callback. For more information, 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. For this, please see Can I connect my contact form with Zendesk?.