I used "Do they or don't they have?" I understand why that is incorrect - it is missing an object. But the accepted answers are missing an object, too, so I agree with wyqtor that adding "it" would be less contrived. A complete sentence would never be "Do they have or don't they have?" This could also be easily solved by adding ellipses, as such: "Do they or don't they have..."
Technically, the exact translation, word for word, would be 'They have it, or don't they?'
While i understand adding 'do' to the front, resulting in 'Do they have it, or don't they?', adding an additional 'have it' at the end is complete nonsense. The word 'have it' simply isn't there in the second part of the sentence.