Some vaccinations are mandatory anyway in germany
MMR, Measles, Mumps, and Rubella, is a combination vaccine that is mandatory for children to attend any kindergarden or school in Germany. Since homeschooling is not allowed (you need to attend a public or private school that is registered with the education ministry), this effectively removes the choice of not getting vaccinated for MMR from parents under the Masernschutzgesetz/IFSG § 20. The only legal exception to not receive MMR-Vaccination and attend a public school would be medical, never religious or other. In case the parents can't put their stuff together and get the child the mandatory vaccination, the court can order the child inspected by a medical professional who is allowed to testify in court and familiar with the case. Usually, that will be done by an Amtsarzt, who is employed by the health office, who in turn requested the medical history from the Hausarzt1. In extreme cases, the court can order vaccination "von Amts wegen" (by the [publich health] office).
Parents need to parent together or the court will decide.
As long as the parents both have medical custody ("Gesundheitssorge"), both have to agree on big parenting decisions, including medical decisions for the child. As the BGH decided, big decisions includes vaccination. In case the parents can't agree, the court can and will decide who is right as an arbiter under BG § 1628 and § 1697 a BGB. The court is in those decisions forced to take the position, that is most beneficial to the child, and can disagree with both parents. What is most beneficial is determined by following the recommendations of the permanent vaccination commission (Ständige Impfkommision STIKO) of the Robert Koch Institute (RKI) and a medical examination by the Amtsarzt.
The court can, as a result of such a case, put sole Gesundheitssorge (medical custody) for vaccinations or more with one of the parents - the other parent can, as a result, not order the doctors something in this regard after that, the parent with medical custody can get the child vaccinated, usually following the suggestions of the STIKO. Most on the point: the BGH had a case about this in 2017, resulting in exactly that.
For Example, if Alice wants the child immunized with "liquid Starlight" against "becoming an Orc" and Bob wants it vaccinated against Smallpox. The court can tell them both to step down, the child should get the proposed vaccinations of the STIKO in accordance with a medical examination of an Amtsarzt, and neither of the two proposed treatments unless they can be proven to be harmless. If the parents still can't agree after that, the court might even remove medical custody for vaccination from both.
- Hausarzt can't be easily translated due to the specific position it has in the German healthcare system. The Hausarzt is the doctor who is usually a person's first point of contact with the medical care of Germany. If you need basic treatments, a prescription, or see a specialist, you visit the doctor's office of your Hausarzt first. One of the few exceptions are teeth and eyes. Gaining a slip to see a specialist often only requires a call and picking the slip up, such as for a woman's Ob/Gyn treatment (which doesn't need the slip but often enough is requested) or for continuing care with known medical health issues. In turn, the Hausarzt gets at least a letter from the specialty doctors that informs him about complications, treatments or findings. The idea is, that the Hausarzt is your personal first contact and knows your full treatment history at least in part and his file on you will be the most complete but for the teeth. While, as comments mention, this is quite akin to the general practitioner in the UK, The Hausarzt doesn't or shouldn't act as the gatekeeper.