I thought this feedback would have been posted years ago, but it looks like it did not.

It's one of the things I think Zendesk miss the most, and that many of our clients (as Zendesk partner) would like to use : the ability to embed Zendesk forms into their website.

Pretty simple and straightforward use case : 

1. Define a form in the Admin Center (with fields and conditions)
2. (TBD) Define font, colors, design elements - new
3. Generate a code/script to be added to any website to set-up the form that will automatically create a ticket in Support

When I searched in the Community, I found signs that users were looking for it, like here and there.

The Zendesk answer is : you can always use the Web Widget.
True, but (1) it's not the same experience for the customer and (2) the widget may be sunset at some point, the new messenger getting all the new features and sexy stuffs. So setting up the Web Widget now is not really a good option.

The alternative is : use any form and create a ticket via API. Also true, but requires some additional development and research (for webmaster or developers). So it doubles the work for something that seems very basic.

In the end, I strongly thing this would be a very good additional feature.