If Dolores Umbridge wasn't a Death Eater, or closely associated with the Death Eaters, how did she end up with Mad-Eye Moody's magical eye, which she so tastefully affixed to her office door at the Ministry? At the time Moody's eye is discovered by Harry in Deathly Hallows as he searches Umbridge's office, the Death Eaters are running the Ministry. Umbridge clearly had no fear of any repercussions from the Death Eaters for hosting Moody's eye on her door for the entire Ministry and its staff to see on a daily basis.
Mad-Eye Moody was killed during the Battle of the Seven Potters in Deathly Hallows and his body fell to an unknown location somewhere in rural England. The Order wouldn't go back for Moody's body because they didn't know where it was and furthermore it was too dangerous for the Order to go on a body recovery mission when their might be Death Eaters lurking about.
Keeping all this in mind: How did Umbridge come to be in possession of Moody's eye? How did the Death Eaters know to give the eye to Umbridge? Knowing what we do about Umbridge from canon, could we conclude that she was perhaps involved in the planning of the Battle of the Seven Potters? Or is there another explanation as to why the Death Eaters felt compelled to give the eye to Umbridge? This is rather gruesome, but could she have expressed direct interest in owning Moody's eye prior to his death, as in "By the way, if you come across Mad-Eye Moody's magical eye, bring it to me immediately." It's so morbid -- I'm interested in any information beyond "A Death Eater gave it to her." I think that much is already clear. Has J.K. Rowling ever addressed the issue of Dolores Umbridge and Moody's eye? Speculative answers based on known canon or JKR quotes are totally fine.