In general, likely all states prohibit an attorney from working on contingency on criminal law and family law cases. However, once an amount has been set by judgment the prohibition against contingency fee arrangement no longer applies.
If there is one or more orders against your father for paying child support and he piled up arrears to $80,000, an attorney likely is not prohibited from entering into a contingency fee arrangement to collect the arrears. In a contingency fee arrangement, the attorney will collect his fees as a portion of the money he actually gets from the debtor.
Your mother likely is the one to whom your father actually owes the money. Your mother can contact a few attorneys to see if they would agree to a contingency fee arrangement with her. The most important factor for attorneys likely would be whether your father has any assets.
Since you are still a minor, your state's office of child support services likely can help your mother to garnish your father's wages. Has your mother asked for the state's help?
As an attorney licensed in Arizona, I can tell you that attorneys can accept child support COLLECTION cases on a contingency basis. I don't know of many attorneys that would do so, though. I'm not sure, though, why the previous contributor was discussing contingency fee agreements in child support collection cases.
In any event, Arizona statutes changed recently to state that a child support award, whether it is reduced to a judgment or not, never expires. In essence, if your mother failed to act on the child support order until you were 50 years old, she could technically still seek to enforce it. After that long, though, the judge may refuse to. Since you are seventeen, your mom can easily seek the current and past support that was ordered and likely increase the child support amount. You, unfortunately, cannot seek the past support. By the way, he still owes support until you are 18, even if you have graduated.