Is it possible for no one to have custody of a child?

I do not practice in GA but child custody and guardianship law is similar across the country. It is technically not possible for no one to have custody of a child -- it appears from what you've said that the grandmother has physical custody. Once the temporary custody expired, legal custody would have reverted to the mother unless child protective services (CPS) or the courts were involved.

If the child's mother, father and grandmother and (depending on how old she is) the child agree to you getting custody, you would get and fill out guardianship papers, have the mother and father sign; have the grandmother sign if there is any indication she has any legal right to consent and have the child sign (if she is over 12, 14 or 16 -- I don't know the age at which consent of the child for a guardianship is required in GA) - then you file the papers in court and get a judge to award you custody under a guardianship.

If any of the family members with a right to custody or to withhold consent to the guardianship either don't sign or can't be located, it gets more complicated and you'll need a lawyer.