The Moral Argument for God

[Angrboda]

That is correct.
But that is not what your OP says. There, it says
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1. If objective moral values and duties do not exist, then God does not exist.
2. Objective moral values and duties do exist.
3. Therefore, God exists.

Do you think the two are somehow equivalent? The one you just quoted doesn't even attempt to prove that God exists. It only gives a possible proof that God does not exist. I am confuse

Yes, I think the two forms of the argument are equivalent.

By reversing the first premise and using "not", you create the contrapositive form. At least, that's what a few websites say about the subject. Are they wrong?

They aren't wrong, you're using it wrong. Note that the contrapositive of the first line is, "If God exists, then objective moral values and duties exist." Going from there and asserting the premise, "Objective moral values and duties exist," is affirming the consequent, which is the corresponding fallacy to your denying the antecedent of the contrapositive. As many have noted, you screwed it up.