I have a sentence where it feels like it would be necessary to use the double that:
e.g. "I wanted to ask what is that you would require from me?"
Is my intuition correct? If yes, please explain why and dissect the sentence.
Yes, this sentence is incorrect.
While a "that" would make sense in this sentence, it is more common to use "it":
I wanted to ask what it is that you would require from me?
This can also take a different form:
I wanted to ask: what is it that you would require from me?
@SovereignSun also pointed out that you can use the word "there":
I wanted to ask: What is there that you would require from me?
"that" is also correct, but less common in useage:
I wanted to ask: what is that that you would require from me?