I am investigating how clear surveys on a website are, and was wondering if the phrase
“I merely inquired how your site works”
sounds natural among young Americans?
Depending on what you mean by "natural", yes it is (I have heard it said before, and it's completely grammatically correct). As a US English speaker, I wouldn't generally say this, because to me it seems too formal of a phrase. Instead, I would use this:
I only asked how your site works.
Even then, if I used "merely inquired" (and only in formal circumstances), I would add "as to" after "inquired".
I merely inquired as to how your site works.
I also agree with commentator DoWhileNot; I too would use "merely" in a defensive tone.