I don't believe so. Just to purchase it. On the other hand, since I don't think the statute defines "handgun ammunition" this creates all sorts of odd little wrinkles. For example, if an 18-year-old has a Ruger Mark II and a 10/22, he could buy ammo for the latter but not the former. He could, however, be given ammo for either. If he takes a box of ammo he bought to the range along with one he was given, and along with both his firearms, he could shoot ammo from either box in his 10/22, but in the Mark II he would only be allowed to shoot the ammo he was given; shooting ammo from the other box would mean he bought "handgun ammunition" illegally.
If you go to Wal-Mart and buy .22LR ammo, the cash register will ask if it's for a handgun. If so, it will ask "Over 21?" Otherwise it will ask "Over 18?". BTW, if you're buying multiple boxes and want to speed up the purchase process (and you're over 21), say "Yes" to the question the first time; once the system establishes you're over 21, it won't ask the question again. Otherwise it will ask the "For Handgun?" question for each box purchased [it will only ask the "Over 18?" question once].