To be precise, there are many kinds of moral arguments for theism. The question in the title is really talking about what we might call “ontological” or “metaphysical” moral arguments, the kind which claim that we need God in order to have an “ontological foundation” for objective or absolute morality.
People who defend a version of this kind of argument include a veritable “Who’s Who?” of contemporary Christian apologists: C.S. Lewis (see here and here), Alvin Plantinga (see here and here), William Lane Craig, Paul Copan, J.P. Moreland, Randal Rauser, David Baggett, Jerry Walls, Norman Geisler, Frank Turek, Roger Olson, Michael Horner, and so forth.
While there have been many critics who seem to be clueless about how to refute such arguments (see here and here for just two of probably 100+ available examples), there are many other philosophers who understand the arguments perfectly well and–gasp!–actually offer relevant objections. (What a concept!) In my opinion, the two best critics of ontological moral arguments are Erik Wielenberg (see here and here) and Wes Morriston (see here and here). Why, then, do apologists who’ve written on the topic in the last decade continue to ignore Wielenberg and Morriston?
I’m starting to think Ex-Apologist has a great explanation, albeit one he didn’t invent specifically for this topic. In fact, I think he has a great name for this great explanation. In a post entitled, “Proposal for a New Entry in the Philosophical Lexicon,” he calls this behavior “craiging.” Here is how he defines it.
craig, v. (a) to engage in dialectically illegitimate argumentative maneuvering, such as (e.g.) construing an interlocutor as offering a rebutting defeater for P when it’s more charitable to construe them as offering an undercutting defeater for P; (b) to maintain a somewhat positive image of one’s positions in part by choosing not to address, mention, or cite the strongest criticisms of them; (c) to take up, critique, and/or ridicule an uncharitable construal of the theses and arguments of one’s interlocutor.
 Relatedly: to infer or otherwise assume that because a reply fails to rebut P, it also fails to undercut P.
It is (b) which I think applies to contemporary defenders of ontological moral arguments for theism: they simply act as if these critiques don’t exist.