Can I get a public defender for a child custody case?

No. Public Defenders only represent people accused of criminal charges. You should contact your local bar association and see if they have a list of any attorneys who practice custody law who would be willing to represent you at either no cost (pro bono) or for a reduced cost. Also, if you are the parent who has been sued for custody you can see if there is a legal services agency where you are located that may be able to represent you at no cost

There is a constitutional right to an attorney in criminal matters, but not civil matters, such as child custody. It may otherwise be difficult to find an attorney willing to handle the case "pro bono" (for no charge), but you can call your county's bar association and ask. In Philadelphia, you could call Philadelphia Legal Assistance. If you qualify financially, you may get good advice on how to represent yourself from a a paralegal, from an attorney and--in some cases--an attorney would in fact take your case and go to court with you, but there are only so many attorneys.

Public defenders are only appointed in a criminal case. There are limited circumstances where the public defenders are appointed in quasi-criminal cases. However for a child custody case, the courts will not appoint a public defender.

You should call your local bar association and ask for the legal services number. In my area, we have Mid-Penn Legal Services, who help out those with low income. The bar association also has a list of attorneys who work with those of modest means. Public Defenders are for the criminal end of the courthouse, and that's not what you need. Best of luck to you!