There are more permutations and approaches than we can discuss in this forum, but let us consider the basic opinions and guidelines.
Every mitzva has a stage at which a child is considered higi’ah l’chinuch, one who has reached the point at which it is practical to educate him, and his father is then obligated to see to it.[1] Regarding negative commandments, a father should try to prevent his child from doing prohibited acts even at a relatively young age.[2] However, a toddler lacks the pertinent understanding, and a father can allow a toddler to act as he wishes.[3]
Even when one need not stop a child from doing a prohibited act,[4] it is forbidden (according to most opinions, by the Torah[5]) for anyone to feed him non-kosher food or encourage him to do a prohibited act.[6] It is, however, permitted to put the child in a situation in which he may, of his own accord and interest, decide to do something forbidden, although perhaps only if it is forbidden on the Rabbinic level.[7] For example, the gemara[8] tells of one who lost keys to a beit midrash in the public domain. R. Pedat told him to take children to play where the keys were lost, with the hope they would find the keys, play with them, and bring them back.[9] In contrast, the mishna[10] requires one who sees a child extinguishing a fire to tell him to stop. The gemara[11] explains that this refers to a case in which the child was acting on his father’s behalf. The Mishna Berura[12] says that in such a situation, even a child who is not higi’ah l’chinuch should be stopped.
There are some major pertinent machlokot. The Rashba[13] and the Ran[14] say that one may prompt a child to do something that is forbidden only Rabbinically. However, both refer to cases in which the child acts for his own purposes. The Rashba,[15] for example, claims that the case described in the mishna about a child who was stopped from extinguishing a fire involved a Rabbinic violation, and yet the child was stopped because he was acting for the needs of someone else.
In contrast, the Shulchan Aruch[16] is presumed to forbid prompting a child to violate even a Rabbinic prohibition, including when it is for his own purposes. However, many poskim justify relying on the view of the Rashba and Ran, at least in a case of the child’s significant need.[17] Rav Ovadia Yosef[18] rules that in a matter in which there are legitimate opinions permitting an action for an adult and it is at worst a Rabbinic prohibition, all would allow one to prompt a child to perform the action for himself.
Another disputed circumstance is one in which adults need the child to violate a Rabbinic law in order to enable the adults to fulfill a mitzva. There is a rule that one can ask a non-Jew to do something that is Rabbinically forbidden for Jews in order to allow a Jew to fulfill a mitzva or when there is a great need.[19] The Taz[20] says that the same applies to asking a child to violate a Rabbinic prohibition under those circumstances. A famous example is the ruling of R. Akiva Eiger,[21] who allows a child to carry a chumash to shul to read from, in a place where there is no eiruv, when it is an adult who has a real need to also use it there. Yalkut Yosef[22] writes that one does not have to protest against those who rely on this opinion if a non-Jew is not available. A child may even act on behalf of an adult if the latter refrains from some matter just as a personal stringency.[23]
Apparently, there are those who make use of their children on Shabbat more freely than others. As long as they do so in a careful way, ensuring especially that the action does not involve any Torah violations, they have legitimate halachic opinions upon which to rely.