Can 15-year-old move in with her grandmother without legal documents if she has her single mother's permission?

Well, yeah, if Mom is the custodial parent and she gives permission for it, then you can move in with Grandmother. But, if Grandmother does not have some type of 'legal document' such as temporary guardianship, then there are going to be potential difficulties that Grandmother will face. For example--enrolling you in school, consenting to your medical care, etc.