According to the US Bureau of Consular Affairs (click "More" under the parental consent section), your sister will need to provide legal evidence that she has sole custody of the child, or submit form DS-5525, along with evidence for the special circumstances she describes. If she does not have this yet, then she will need to acquire this (shouldn't be especially difficult for a vanished, non-maintenace paying parent).
If the child only has one parent/guardian, evidence of sole authority to apply for the child must be submitted with the application in the form of a:
- Court order granting sole legal custody to the applying parent (unless child’s travel is restricted by that order)
- A court order specifically permitting the applying parent to apply for the child’s passport
- Judicial declaration of incompetence of the non-applying parent
- Death certificate of the non-applying parent
- Photocopies and notarized copies are unacceptable
If the child has two parents/guardians, but one is absent and cannot be located to provide parental consent in a timely manner, the applying parent must submit Form DS-5525: Statement of Exigent/Special Family Circumstances. The statement must explain in detail the non-applying parent's or guardian’s unavailability and recent efforts made to contact the non-applying parent. The applying parent also may be required to provide evidence (e.g., custody order, incarceration order, restraining order) to document his/her claim of exigent or special circumstances. To protect against international parental child abduction, the Passport Agency processing the application may ask for additional details if the statement is determined to be insufficient.
In short, consulting a family lawyer may be useful (especially as the issue could well come up again in non-travel circrumstances, at potentially more inconvenient times.