While it's easy to call emergency services by 911, it's more difficult to call, say, the local sheriff's office for something that's not an emergency especially when you're traveling. An app could help solve that and resolve the best non-emergency number for you in your current location.
Isn't that called a phone directory? I can go to my country's directory website, type "police", choose "near me" (it gets the location from the device) and click search to get the number.
Establishing a non-emergency number is not hard. Dial 101 in the UK and you're put through to your local police force: http://www.police.uk/contact/101/
In the UK we have 101 for non emergency police and 999 (or 112) for emergency. I think 111 does a non emergency call to the health service where they will listen to your problem and give you advice (ambulance, doctor the next day, your fine etc.).