The reimbursement of pupils' travel expenses in Nordrhein-Westfalen depends on the distance to school (the shortest path between home and school): the simple distance to primary school must be more than 2 km, in secondary school and in class 10 at the gymnasium more 3.5 km and at secondary level II more than 5 km (https://www.schulministerium.nrw.de/docs/Recht/Schulrecht/FAQ-Schulrecht/FAQ-Schulrecht-Eltern/FAQ-Schuelerfahrkosten/index).
If your child is entitled to benefits from the education and participation package, the costs for student transport between the place of residence and the school will be covered (https://www.schulministerium.nrw.de/docs/FamilieBildung/Bildungs-und-Teilhabepaket/index.html).