Moth balls? Not a good idea. They are poisonous. If the stray cat touches it and then licks its paws later, it will be poisoned by it. Is there not a rescue group in your area who will step in and trap, spay and rehome this cat? Check the link below. Click on the link, your State, and each of the numbers to find a rescue group near you( if the one you spoke of doesn't want to get involved-which it doesn't sound like it if they left it up to you to do the trapping).
Hugs for Homeless Animals
www.h4ha.org/shelters/
A worldwide directory of rescue organizations, shelters and humane societies.