He’d never said anything apart from the customary, “Hello(,)” and “May I take your order?”
Do I need that comma? Why or why not?
You don't need the comma.
You are listing two phrases that are customary in the current context. You're wrongly trying to add a comma here, because to your mind, these two things were said one after the other (as a single sentence).
However, the current statement you are making does not present these two things as a single sentence. It lists them as two separate things that are customary to say. And therefore, no comma is needed.
He’d never said anything apart from the customary “Hello” and “Goodbye”.
Now that I'm listing two things that would (presumedly) never be said in succession, I suspect you agree that there is no need for the comma here.
But the principle is the same whether or not this person said these two things one after the other or not.
The only reason to use a comma is if you are stating this as a single sentence:
He’d never said anything apart from the customary “Hello, may I take your order?”
Here, you can add the comma, because it is part of the full sentence that the person says out loud. This is not a list of two separate things to say, it is a single one.
Notice in my above examples that I removed the first comma too. It does not belong there. You do not need a pause in the sentence at that point.
This is the most acceptable place to put a comma:
He’d never said anything , apart from the customary “Hello” and “May I take your order?”
Even this is highly optional. I would add it only if it represents a pause that was taken by the speaker.
e.g. the speaker only then realized that he actually did say some customary things; so they corrected their earlier statement in which they claimed that he had never said anything to them.
There is no comma needed for grammatical purposes anywhere in the sentence.