I agree.  Keep a log of when you called, what number you called, whose extension (if any) you were tranferred to, etc...

Provide this info to the client and give them as many contact options as you can.  Include phone numbers, email (if any), and street/mailing address of the agency, plus their operational hours if you have that information.

Tell them that you've gone to extraordinary  measures and beyond reasonable diligence in attempting to collect information, and that the public agency has not yet replied or been available on their public phone lines or otherwise.