Based on my own experience while moving cities within Germany, and assuming it's a public or kath/ev KiTa the child is at, it's a trifecta that usually gets the kid into school before it is supposed to:
1. The kindergarten teacher has to agree, taking in consideration the development of the child, its abilities, results of required standardized tests (e.g. Delfin 4), etc. This should be recorded all in the child's file. 2. The Kinderarzt has to agree, because the child might have physical or developmental issues that might hinder its education if schooled earlier. 3. When the child is placed in Vorschulkind status, the Jugendamt also has to agree. They mandate another exam by an official doctor. It's physical in part, and in part examines the child's ability in various tasks (for which usually the child is indirectly prepared in the KiTa). If no issues arise, your child is deemed Schulfähig.
Next step would be to start visiting primary schools. I believe all of them has a sort of open door day where you get to see the school and get the info regarding registering the child for admittance.
Then again, better take official advice from the burocrats themselves, rather than strangers online: