One quick correction is that elephants were native to Iraq in the historic period, as Syrian Elephants inhabited the region until some point in the first millennium BC. But obviously Saddam wasn't time-traveling to collect local elephants.

As for the claim, this is one of those things that is hard to track down. I see that wikipedia talks about it as the last "recorded" use of elephants, but once I start digging into sources this becomes "alleged". A lot of the claim for Wikipedia seems to be lifted from The Last Giants by Levison Wood, who says the last "supposed" use of elephants was in 1987 when Iraq used some to transport heavy weaponry to Kirkuk. The time and place would make it sound like something connected to the domestic Al-Anfal Campaign against Iraqi Kurds...assuming it actually happened. Wood doesn't actually cite a source, regardless.

Much of the rest of the internet claims for this instance cite Jared Eglan's Beasts of War: The Militarization of Animals. Unfortunately here, I can't find a digital copy to check. There's not much information about Eglan, either (this seems to be his only book), and in any case it's described as a "curated collection of articles s and information about the militarization of animals throughout history," so I'm not even sure how much of it is actual original writing or research (it looks to be self-published).

Other than this I'm not coming up with anything. A book on the history of war elephants, appropriately titled War Elephants by John Kistler, discusses use of elephants in Iraq in historic times, but makes zero mention of modern era use.

As for elephants in Iraq during Saddam's time - there should have been a few, although I'm again having trouble tracking down exact numbers. Lawrence Anthony, in Babylon's Ark, writes about trying to save animals and reconstruct the Baghdad Zoo after the 2003 invasion - he mentions an elephant enclosure, but it seems like the elephants (along with most of the prewar animals) were long gone. There also was a zoo in Mosul before 2003 (which is much closer to Kirkuk by the way) as well as a number of private menageries owned by Saddam or his relatives.

So: this is not the final word (I don't read Arabic and someone who can check Arabic sources maybe could get deeper insight). But this sounds very much like a story of something that might have happened in 1987, but if it actually happened looks like it was for a specific instance of transporting something within Iraq to Kirkuk, and probably involved using local elephants for the purpose. But it's really hard to pin down whether this actually happened, and seems to have turned into a history factoid (similar to the whole Pepsi Navy thing) through a game of Internet Telephone. It's not like Iraq had a War Elephants Corps that they were operating on a regular basis until 1987.