What is the best way to construct this sentence? Basically, there is a question in the middle of the sentence.
"Clutching the envelope in her hand, who sent letters by certified mail anymore, anyway? She hunted through her cluttered desk for her rarely used letter opener."
In normal sentence construction, I would need a comma after the "anyway," yet, since this is a question, I can't have a comma after a question mark. What is the best construxtion for this pair?
The question mark is fine as is. That's not a question in the middle of a sentence. That's two sentences. The question mark ends the first sentence. If it weren't a question, you'd put a period there, not a comma.
The thing is, that first sentence has another problem that may be what's throwing you off. It has a dangling modifier. Who is clutching the envelope? "She", I assume. When you start a sentence with something like "clutching the envelope" you have to write the next part so that the same person is doing both things. The way you wrote it, "she" is the one doing the clutching, but "who" is the one doing the sending.
There are several ways to fix it. Here are a couple of possibilities. One is to change the "who" part to make it something "she" does. I added the words "she wondered". A side effect of that is that it isn't a question anymore.
Clutching the envelope in her hand, she wondered who sent letters by certified mail anymore, anyway. She hunted through her cluttered desk for her rarely used letter opener.
Another choice would be to rearrange the order and attach the "clutching" part to the second sentence, which already starts with "she".
Who sent letters by certified mail anymore, anyway? Clutching the envelope in her hand, she hunted through her cluttered desk for her rarely used letter opener.
In both cases, notice the relationship of the words I bolded.