We have an injured black vulture (wing; can't fly) on our property. Is that something animal control would manage?

No

There used to be a vet in Florence. Call a veterinarian and ask.

they typically deal with dogs and cats, but it couldn't hurt to call and ask. They should have someone on staff at all times for the animals, but they may not answer during non business hours. If that happens, try calling the police non emergency number and they should have an answer.

I am unsure but I dont think so. They may refer you to a no cost option