My cat is friends with a stray

This stray is my old neighbors cat Chevy. When they moved, they intended to have Chevy put down because she has FIDs and they also suspect cancer, but she ran away and they couldn't find her again before they left. She is skinny, missing fur, walking funny, and shouts at you when you look at her. (I feel "meows" doesn't accurately describe the noise she makes.) Chevy is also...rather friendly. 

I keep finding Chevy on the porch with my cat Kiki. They've been napping together and I'm pretty sure Kiki is giving Chevy her food. I keep thinking about calling animal control to come get Chevy, but they're open at weird hours and now the more I think about it...I don't think Chevy can get Kiki sick. They're both girls and I don't believe they will fight. If animal control comes and gets her, they won't get her help they'll just put her down. I can't afford to take her to the vet, but I can feed her. 

Is it better to leave her be, and let her have a friend for the limited time she has left, or have her put to sleep so she's not suffering through this anymore? Is she a danger to my cat? 

This is tricky. 

I think I have to say that it would be most humane to put her down if you're not able to take care of her veterinary needs. Otherwise, she'll keep suffering and her death will be slow. Furthermore, even though she hopefully won't infect your cat, there's a possibility that she'll infect other cats in the area. This disease is very important to subdue and prevent, segregate and treat those that are infected. Unfortunately, it doesn't seem like you're able to do this, so the best option would be to give her up to a shelter. She'll probably be put to sleep, which I know is awful, but it's the most humane option.

Maybe you can find a no-kill shelter/organization that can help you? 

The closest one is an hour away and is not accepting any more animals, and that's about it. I'm not even sure the local shelter would take her, considering they're always full on cats.

Can you take her in?  That would be a good solution if possible.

Well for one, I can't afford to get her the veterinary help she needs, and even if I could, I can't get her to the vet(I had to ask a good friend to drive me to the vet when Kiki had an infected wound, my family refused saying it wasn't important enough), further I don't even think I could catch her. My family doesn't like me  having the cats I have, if I attempted to take in another they would put their foot down. They might even shoot her. 

I'm not sure what to say about catching her other than using a humane trap. As far as a ride to the vet, I don't know what to say about that at all. I'm sorry your family isn't more supportive. Regarding paying for vet treatment, the group we foster for helps people with veterinary costs. They have to take the animal to our vet where we have an account. You might ask some of the local groups if they can help catch her and get her vetted. They can treat her as a TNR.

Are you anywhere near Clanton? I know someone there and could see if she has any ideas.