What can I do?

Hi, my name is Michelle, recently we got new neighbors and they have a cat about 1 year old, they leave her run freely around our apartment building, she has no collar, she comes over to visit me when I go out on the patio I also give her food evey morning she is just waiting for me to get up to give her food! I feel so sorry for her she is white with one green eye and one blue eye! and she is dirty as can be, her ears are filthy inside and yesterday I noticed she has a bit of blood coming out of one eye, and today I seen she has a gash by her tail like something bit her or maybe got hit with a bb gun? Honestly I wish I could just take her in and give her the care she deserves, she is as sweet as can be, I feel so angry that these people dont do anything, what can I do?

Welcome to TCS, Michelle! you are this cat's guardian angel - i am so glad that she found you, cuz it sounds like she's on her own, basically. Your neighbors definitely need a "humane education" course Do you have a humane society (the ASPCA or HSUS) in your area? If so, call them for advice.
As for your new neighbors, it would help if you can develop a good repoire with them, so they might be more open to any valuable advice that you can give them.
If it were me, I would wipe her with a damp washcloth and put some neosporin on her gash. i don't know if you could afford to take her to the vet - i don't think the vet could legally treat her if she's not yours, but proof of ownership is difficult. i think that since her injuries are not life threatening, you will have to go thru the owners.
please keep us updated on the little cutie & how she's doing! I'm sending {{{protection prayers and vibes}}} to your new kitty friend! Susan

Seriously if it were me, your neighbors are obviously too stupid to own a cat. It is not up to me to tell them they can or cant have one but for the sake of the cat, if you are willing and able to take this cat to the vet and get her looked at (check if she is spayed too and if you can afford that...) then I say go for it. You are making this cat better by taking it in. Do the neighbors know you or know you know it is their cat? Can you get away with saying you had no idea it was theirs if they ever say something to you about it?

I am not saying to steal their cat, just get it the medical attention including possibly spaying and vaccinating it, then either let her go or go take her to the neighbors and play dumb and ask if it is theirs. Inform them that she was injured and you took her to the vet and anything else you may have done. Then go by their reaction. If they seem to not care ask if you can keep her, then either keep her or rehome her whatever). Never know, maybe they will be greatful and repay you.

One possibility is also, as you ARE prepared to help this cat including paying vet; you go to the neighbours, and say you get quite fond in the cat; could you have it for your own please??

Chance is good as they essentially dont care for the cat, they will only be happy if she gets a now home.
Only danger they will charge you to buy the cat... You can riposte she you must take her to the vet, get shots and so on...

An alternative may be you go to the neigbours, and offer you can help them with getting the cat to the vet. And cat sitting should they need.
Some people arent evil by themselves, they are only slow with getting things done, no enterprise in them. Here: Themselves will be slow with getting to vet, but will appreciate to get help to get the cat to vet, and may even pay for the vet - hey, may even pay YOU for the help with getting the cat to vet...
There are different possibilities... You will see if you talk with them and have your ears and eyes wide open.

Good luck!