It doesn't seem to fit in with the meaning set out in my dictionary.
In my dictionary it says the following.
For familiar language it marks wh-question (what, where, who).
For children's language it means "strengthens one's judgment or conclusion".
Because it is only for questions of what, where, who.
If I understand the sentence correctly, it means "How the heck is one going to support the family?"
So the question is how どうやって, not what, not where, not who.