"The journalist does not want the lawyer to write the story."
Translation:Journalisten vil ikke have, at advokaten skal skrive historien.
Technically: Yes In practise: No I believe leaving 'have' out would technically count as correct Danish, but you you would sound weirdly formal and old-timey.
This is not a literal translation, as we see in Duolingo so often. Maybe it would be better to change either the Danish or English sentence slightly? For example: "The journalist does not want that the lawyer will write the story". That would reflect the Danish sentence better.
But it's awkward English. (And I'm not sure that it does reflect the Danish better. Take this with a grain of salt, as I'm no expert, but I think the skal here shows more intent than plain future tense does.)