at http://www.wachau-blog.at/gutscheine/ some fields are styled with overflow:hidden by your css and thus are not displayed (Name, Anschrift).
Why is this - did I miss a setting?
The overflow rule is used to clear floated elements without adding additional markup. There is no setting, this is purely part of the CSS.
It works fine normally and 99% of users don't have a problem with it. The form is inheriting something from your theme that's causing the issue.
If I toggle your theme styles off, you'll notice that the fields display properly with the default form stlyles.
I'm sure if you check in your stylesheet, you'll find some additional margins, padding or positioning rules that are causing the fields to be displayed outside of the overflow area.