In short, the sentence is not completely said. 会えなく is the 連用形 of 会えない (なく is the 連用形 of ない), and it marks a mid-stop in a sentence, aka the sentence is only half said at this point.
In English, you can observe the same feature:
If I couldn't meet you...(sentence not complete)
もしあなたと会えなく...(sentence not complete)
It tells you she didn't say completely what she wanted to say, but the author didn't write the other half of the sentence, so the readers have to infer.
In modern Japanese, one would usually connect two clauses with te-form, and the use of 連用形 is somewhat from Classical Japanese, and when people use it, it gives more formality than the te-form. As shown by this picture I found (source: https://honda-n2.com/kobun-kobuntangochou-osusume)
The 連用形 of the verb 降る is 降り, but the 連用形 for ない is なく. Of course, as shown in the picture, we can replace 降り with the modern te-form 降って, and for your sentence, it would mean the same thing if we use the te-form
Maybe it conveys less formality but in this context I don't think it matters, since this does not seem like a formal situation at all.
I have found this additional info on Japan Reference that I think could be helpful. Here is the link to the post, but here's a screen shot in case you want to know whose answer I'm talking about.