The shulchan Aruch (657) writes that a father should fulfill the mitzvah of chinuch and buy his son a lulav if he is old enough to do na'anuim. The Maharashal questions the need for the father to purchase a lulav. The mitzvah of lulav entails taking the lulav, but there is no reason a child cannot fulfill the mitzvah of netilas lulav with his father's lulav if he does not own his own. It sounds like there are overtones of yesterday's hechsher mitzvah discussion here as well. It is unclear to me if the SA's position is based on 1) a simple blurring of the lines between buying a lulav and the mitzvah of netilas lulav because practically speaking it is easier to do na'anuim (certainly during hallel, see Taz) if you own your own lulav, 2) the assumption that there is a
chiyuv of chinuch even for a hechsher mitzvah, or 3) somehow the purchase of a lulav is considered more a part of the mitzvah of lulav than a regular hechsher mitzvah.