Adverbs describe the manner in which something happens. In your example sentence, there is no manner being described. Rather, the sentence describes a transition into a particular state of being. (After a number of comments, perhaps it's just best to jump to the end of this post where I walk through the traditional Japanese grammatical way of thinking about this. Japanese is not like Indo-European languages, and so at some point, clinging to Indo-European grammatical categories can get in the way.)
Thinking from an Indo-European Perspective
I would be careful about how deeply you try to parse the grammatical forms. Form and function are two different things. It's important to keep them straight (particularly when using non-native terminology to talk about the grammar).
In English, you can say adverbs end in ly. Does that mean that bully is an adverb or that silly is an adverb? But, fully is indeed an adverb.
Analyzing 会わなく as an adverb is not going to help you understand the Japanese any better. 会わない is in form an i-adjective, as such it inflects much like an i-adjective does.
If I want to say, My apple is red, you would say.
りんごがあかい
If you wanted to say My apple is turning red, you would say,
りんごがあかくなる
This does not say, My apple is becoming redly.
会わない is how you say "they don't meet". If you want to say, "they've stopped meeting", then one way to say this is to use なる.
But the grammar of なる requires a particular form precede it.
If it's a noun, then になる
カエルになる
He turned into a frog.
If it's an i-adjective, you replace い with く
あたたかくなる
It'll get warm.
No one understands these as "He becomes froggily" or "It'll get warmly".
Yes, in form these could be construed as adverbs, but they don't actually function that way in the sentence; it's just part of the form into which the word must inflect to function properly in the sentence.
I can say
去年ほんを毎日読んだが、さいきんほんを読まなくなっている
Last year I read books everyday but recently I've stopped reading books.
The last half of this sentence is not saying
I've become in the manner of not reading books
As convoluted as that sounds in English is just how convoluted it would be for an individual to understand 読まなく as an adverb in Japanese.
In summary
An adverb describes the manner in which something happens. But in these example above, we're not describing the manner in which something happened but rather that state into which things transitioned.
Likewise, Riko and Mai transitioned to a state where they were no longer meeting each other. They did not become not-meetingly.
ADDENDUM: Thinking from a native Japanese grammatical perspective
The construction that you have is as follows
あう ＋ ない ＋ なる
to link these together, you must sequentially construct the [連用形]{れんようけい} of each component--this not entirely true, please read on.
The 連用形 of あう is あい
The 連用形 of ない is なく
Except that ない does not attach to the 連用形; it requires the [未然形]{みぜんけい}. Which, in the case of あう is あわ
Now you can take
あう ＋ ない ＋ なる
and modify the forms so that they can be glued together
あわ＋なく＋なる
and thus you get
あわなくなる
Perhaps the key here to help you understand what's going on is to remember (or discover if you didn't already know) that Japanese is largely an agglutinative language. That means (pardon me if you already understand this much) that words and grammatical structures are essentially built up by stacking them on top of each other. But before you can stack them, they must be tweaked into the proper form to match the requirements of the parts that are being glued together.
