My little sister is going to be 16 next month. Her father and our mother had recently gotten divorced. They didn't go to court or anything, just filed through a lawyer. There were no legal documents of any sort of custodial rules or guidelines. She lives with her father, but she wants to move in with our mother. Her dad, however, won't let her do so. She has to fight with him just to get him to allow her to see my mother on weekends. Is it not her right to decide which parent she wants to live with without any problems or court-related situations?

If they got divorced, one of them at least had to appear in court. And part of a divorce is custody agreements.
Are you sure they are really divorced ?

But this is up to the mother, if there really is no custody agreements, both parents have equal rights to have the child live with them.
So it is up to the mother to act and do something

First, she has no right to decide anything regarding custody. She is a minor.

Second, how did they manage to get divorced with no decree granted by a judge (court appearance) and no determination of custody? Some information is missing here.