When people ask for pet food, they get a form where they can request spay / neuter or other veterinary care assistance, pet behavior counseling and other resources. (animalsheltering.org)
PALS: Pets Are Loving Support (veterinary care assistance programs for senior, disabled or ill pet owners) (humanesociety.org)
There are also special programs for veterinary care assistance for working dogs and service animals, such as Assistance Dogs Special Allowance Program and The Gandalf Fund. (animalwelfaresociety.org)