This is a sentential relative clause, which is a special kind of relative clause that can be described as a disjunct adverbial.
Even the popular sources for online grammar help are wary enough of this construction that they allow exceptions to modifying noun phrases in their definition of relative clauses. For example, here's ThoughtCo:
A relative clause is a clause that usually modifies a noun or noun phrase and is introduced by a relative pronoun (which, that, who, whom, whose), a relative adverb (where, when, why), or a zero relative.
One exception to usual protocol is the situation you're raising, where the relative clause seems to modify the entire statement rather than a single part of it. In your example,
My son chews with his mouth open, which aggravates me.
An initial reading might treat which aggravates me as a nonrestrictive relative clause, meaning that removing it does not restrict or limit the sentence it modifies. That would work, except I've changed the definition of nonrestrictive relative clause a little bit: many sources will insist that a nonrestrictive clause does not restrict the noun or noun phrase it modifies (see ThoughtCo again). So this kind of clause would require a definition that goes outside of the "restrictive / nonrestrictive" distinction.
This logic leads some guides, like The Oxford Dictionary of English Grammar, to specify a sentential relative clause under "relative":
A third type of relative clause (though treated as non-defining in some models) is the sentential relative clause, which refers back to the whole or to a part of a previous clause. It is usually introduced by which: "The hotel is very expensive. Which is a pity."
English Grammar Today provides some corroboration for this usage (I have modified the formatting for the example for readability on this site):
The other type of relative clause refers to a whole sentence or stretch of language (they are sometimes called sentential relative clauses). This type of relative clause is always introduced with which. In writing we usually put a comma before which:
But I think Sean was a bit upset about that, which is understandable. (which is understandable refers to the whole clause before it [italics]: that Sean was upset about something)
Your central question is whether this functions adverbially. Yes. English linguist Hilde Hasselgård describes the phrase as a "disjunct adverbial," meaning that it comments back upon the clause it is modifying. Lecturer in modern languages G. David Morley makes the same argument in Syntax in Functional Grammar: he refers to the following example as both an adverbial clause and a sentential relative because it "refer[s] back to the whole of the antecedent main clause":
We still haven't had an official reply, which doesn't help matters.
The same is true of your own example, as mentioned earlier. It's an adverbial.