Can a mentally and physically disabled child be wheeled in a wheelchair on Shabbat in a place without a kosher Eruv? What about if the Goy does it?

It is permissible for the child to wheel himself, if he is capable of it, in a place where there is no Eruv, but there must be nothing in his wheelchair other than the pillow he sits on or in his pockets. A gentile may be told to push the wheelchair (empty of everything except what is necessary for the travel of the child) to shul or elsewhere for the sake of a Mitzva- so that the child will eat the Shabbat meal, hear a Shiur etc. (Shmirat Shabbat K’Hilchata I, 34, 27).