In what follows I will try to outline a basic argument for why intelligent design may be a naturalistic explanation, at least if one is willing to accept that something like the big-bang episode is a naturalistic explanation.
First, something qualifies as a naturalistic explanation just in case it is empirically verifiable and proposing it does not commit one to any supernatural cause or causal agency.
It is not necessary to say that something is disqualified from being a naturalistic explanation if its cause is actually not natural, since in the case of the Big Bang episode one the actual cause may just be God. One is not scientifically committing themselves to God once they admit the Big Bang, first because that is outside of the scope of science in principle, but also because there is an alternative: the Multi-verse. Of course, somebody who believes the Big Bang is a scientific explanation is also not committing themselves to anything like the multi-verse. In fact, there may be good reasons to think that no naturalistic causal account of the universe is even possible (the multi-verse will not even push the issue back a step if one simply means by the universe the whole of nature, and by the big bang some finite beginning to it – and the Borde Guth Vilenkin theorem proves that a multi-verse system requires something like its own big-bang episode). Therefore good explanations, themselves otherwise naturalistic, might commit us philosophically to super-natural causes without committing us scientifically to them. Moreover, science leaves us free to do good philosophy without pretending that science can ever do the job of philosophy, and thus science may license the claim that supernatural causes can be reasonably inferred without science committing itself to such supernatural causes. This is why it is reasonable, for instance, to have arguments over why the universe is intelligible at all and why science works, and to propose God as the best philosophical explanation for the curious usefulness of science. That argument may or may not be good, but it is obviously not for science to decide, but for philosophy. With all this in view, it becomes clear that the Big Bang is closer to implying super-naturalism than Intelligent Design.
Now, in the case of intelligent design, it seems obvious that some kinds of intelligent design, such as Michael Behe’s proposal of irreducible complexity, satisfy our definition above of a naturalistic explanation. It is empirically verifiable and falsifiable, and it does not commit one to any supernatural causal agency, as it is logically possible that some alien species evolved in a series of successive steps gradually (such that at no step was a machine produced which would qualify as irreducibly complex), and then seeded life here by design, or oversaw certain evolutionary steps and imposed design.