Kudos to you for wanting to protect your sister. Family court laws and norms are different in every state. I live in Texas, and I'm not a lawyer or social worker, so please don't take any of this as gospel.
You can file a complaint with CPS based on what your sister told you. They should investigate, and they will hopefully be able to tell you the results of that investigation.
Keep good documentation - write down everything that your sister tells you about how she is treated when with her mom and with your dad. Make a note - if you can - of when she sees her mom and when she is staying at places that aren't with your dad. Write down every bruise you see on her, any change of attitude that you see in her/any signs of depression, etc. Always note the date and time and how you came by the information ("I saw it"/"my sister said..."/"my dad said....") If, when school starts, her grades start slipping or she's not doing her homework, etc...write it all down. Even if you don't use this information for some time, you are documenting a potential pattern.
Is she your dad's biological child? Does he have a legal custody agreement with her mom, or are they just making it up as they go along? Are they legally separated or just informally?
Custody orders are never "permanent" as in set-in-stone-until-the-kid-turns-18. They are "permanent" until someone files for a modification. If no one files for a modification, the order stays in place. If someone does file, then the court rules on the merits of that request/orders mediation/etc. [My husband has been divorced for 9 years; he's recently filed for a modification to change the terms of the custody agreement with his xW.]
Family court focuses on "the best interests of the child". Usually, the status quo is kept unless evidence shows that the status quo is not in the best interests of the child, as defined by your state. (Hence, documentation!) This means that it isn't really necessary to prove a parent is "unfit" - it's necessary to prove that the parent's ACTIONS are harming the child's physical or emotional well-being. CPS uses "unfit" to determine whether or not to revoke parental rights completely.
"Joint custody" usually refers to decision-making power. It means that both parties have equal rights to make medical decisions, be involved at school, etc. Usually one party is designated as "primary", so they have the right to set where the child goes to school.
"'Posession" refers to who the child lives with when. This is the visitation schedule. It is possible to not have joint custody and still see the child a great deal, depending on the terms of the agreement/judge's order.
If you believe that your sister is being abused and your dad is not taking action to protect her, is he the right person to be in charge of her? Does he not believe her, or just believe that she needs to see her mother, or just want a break from caring for a kid for the summer/weekends? Would she be better off in general living with you?
In your shoes, I would contact CPS immediately and tell them everything that you know. Make sure they also know that your dad has heard these allegations and done nothing. If CPS finds the allegations credible, they will also investigate your dad. They can keep your identity a secret...but your dad may still figure out that you were the one to call.
When school starts again, you could also call the school counselor and her teacher and let them know what is going on and ask them to watch her for signs of abuse or distress. Because of privacy laws, they probably can't tell you what they think, but they are mandatory reporters and would have to tell CPS if they see signs of abuse.
At the same time, I would ask your dad if your sister can spend more time with you. If he does agree, this would start establishing a status quo. If he doesn't agree, keep asking if you can have her for certain weekend. Also make sure that your sister has a way to contact you and feels safe in confiding in you.
Here's the pure speculation -
If your dad is the bio father, I can't really see any way that you could get joint custody. If your dad and the mother aren't legally separated (it's informal), then I can't see any way that you could get joint custody. If it comes down to it, you will likely need to petition for full custody...and that means you will need lots of evidence to prove that both her mother and your dad are not acting in her best interests and that you can be a better parental figure.
Best-case scenario - both her mom and your dad agree that you would be better able to raise her and agree to sign papers making you her legal guardian and giving them some form of visitation. [This is likely what is going to happen in my husband's custody modification - now that he's shown stepdaughter's mom the reasons why he wants to make the change, I think she will agree to what he is asking.]
If either of them disagrees and wants to fight that change, this could be a very long and expensive battle. It could irrevocably damage your relationship with your father (because you would be telling the court that he isn't fit to make these decisions), and you are not guaranteed to win - it's an uphill battle when you aren't one of the parents.
Your sister is lucky that she has you trying to protect her. I wish you the best of luck in finding the right path to do so - whether it's being a safe space to listen and show her a different kind of life..or whether it's the extreme of filing for custody.