However, the Torah prohibition of carrying was almost inapplicable in the Maharam Shick’s time, since there were no roads commonly traveled by 600,000 people. Therefore, perhaps small children may be asked to carry for their parents (Teshuvos Maharam Shik, 173).
The Shulchan Aruch’s ruling: This question is relevant only according to the Rashba, who rules that there is no general prohibition against encouraging young children to transgress Rabbinic prohibitions. However, the Shulchan Aruch (O.C. 343) follows the opinion that even this is forbidden. Therefore, there is certainly no room to permit asking a child to carry for an adult, even in a Rabbinically forbidden area.
Chinuch: Furthermore, this discussion concerns only children who are too young to be educated in mitzva observance. Nevertheless, our sugya forbids feeding them Torah prohibited foods. Here, the Rashba rules that Rabbinically prohibited foods may be fed to them. However, the Rashba certainly agrees that once a child has reached the age to be educated in mitzva observance, his father must teach him to observe all the mitzvos – both Torah and Rabbinic. Therefore, he may not carry a siddur even for himself (see Mishna Berura 343 s.k. 3). R’ Akiva Eiger seems to apply that the obligation of chinuch in mitzva observance begins at age nine. The Maharam Shick suggests that perhaps asking a child to carry a siddur to shul is also considered good chinuch, even if there is no eiruv, since one is educating him to daven.