There is no law requiring administration of a rabies vaccines, instead it is just highly recommended for certain kinds of travel. Normally, there is nothing to refuse. However, in case of exposure, there is a course of treatment that includes, in part, administering the rabies vaccine on days 0, 3, 7, and 14. The CDC also says that it usually takes 24 to 72 hours to confirm that a suspected animal has rabies. Therefore, the fact of developing rabies is highly uncertain, and can be complicated by the legal status of pet cats, dogs and ferrets versus wild / feral animals. There is a traditional 3-day rule for treatment in case the animal is captured, otherwise there is no way to tell. Waiting for symptoms to develop in the bite-victim not totally pointless, but apparently "wait and see" is still a bad idea. It is also reported that the treatment is not fun.
Children are deemed to be legally incompetent to make their own medical decisions, thus the parent uses their judgment, acting in the child's best interest. The law respects those decisions except when they place the child’s health, well-being, or life in jeopardy (Meyer v. Nebraska, US 262: 390 (1923), Pierce v. Society of Sisters, U.S. 268: 510 (1925), Wisconsin v. Yoder, US 406: 205 (1972), Parham v. J.R.,U.S. 442: 584 (1979), In Re Phillip B, Cal Ct. App 156: Cal Rptr 2d 48 (1979)).
The most extreme legal response would be prosecuting the parent(s) for refusing to treat the child. In Washington, the statute covering child abuse does not include criminal penalties for refusing medical treatment. However, a hospital administrator can seek a court order to compel treatment, if the court finds that that would be in the best interest of the child. There is a separate crime of criminal mistreatment which could be invoked. The law starts with a religious exception that
a person who, in good faith, is furnished Christian Science treatment
by a duly accredited Christian Science practitioner in lieu of medical
care is not considered deprived of medically necessary health care or
Internet wingnut beliefs are not included. The term "Basic necessities of life" includes medically necessary health care including health-related treatment. The stiffest penalty is first degree criminal mistreatment, when
A parent of a child... is guilty of criminal mistreatment in the first degree if he or she with criminal negligence, as defined in RCW 9A.08.010, causes great bodily harm to a child or dependent person by withholding any of the basic necessities of life, that is, if he "fails to be aware of a substantial risk that a wrongful act may occur and his or her failure to be aware of such substantial risk constitutes a gross deviation from the standard of care that a reasonable person would exercise in the same situation".
If the child contracts rabies and dies, great bodily harm has ensued, but if it's just a skin wound, there is no bodily harm caused by the parent (first degree mistreatment requires actual great bodily harm). Moving down the list, second degree mistreatment is an action which "creates an imminent and substantial risk of death or great bodily harm", or "substantial bodily harm" in the case of third degree mistreatment – both also requiring "criminal negligence".
The crucial question for criminal prosecution is whether the act was with criminal negligence. Here is the instruction that tells the jury how to decide – the instruction is not particularly informative. I tend to think that refusing treatment is a gross deviation from the standard of care that a reasonable person would exercise, but I understand that there are real reasons to not get treated, and also crazy reasons. The law has no explicit provisions that allow you to withhold treatment if you believe that would be in the child's best interest (for instance, if you have a good-faith belief that there is little actual risk that the child has been infected). I have no idea (at the moment) how such criminal cases have played out in Washington.