Appleseed rifle...
Ghost Dog--
ANY working autoloading rifle with a 10-round removable magazine will do perfectly. The Ruger 10/22 is common as eggs, and, fitted with a Tech-sight and sling swivels, it works very well at an Appleseed. You want 2 10-round magazines for it--and I'd bring a third as a spare in case of malfunction. One reason for the 10/22's popularity is that there are so many of them in use, that there is a whole raft of retro-fitting items available for them, so they are very easy to customize. 10/22's are NOT the greatest autoloading .22 available, just the commonest.
But the rule at an Appleseed is, "Bring what you got, shoot what you brung." A bolt-action with a 10-round magazine will work nearly as well as an autoloader. A center-fire rifle will do quite as well as a .22, it'll just be more expensive to feed. A tubular magazine rifle will work; it's just a little awkward but definitely do-able. A rifle without adjustable sights will work; you merely have to use "Kentucky windage." A rifle without sling swivels will work, but obviously, you don't get the (rather large) benefit of using a sling--rests and bipods are a no-no at an Appleseed. A sigle-shot rifle will work, however the timed shoots will be nearly impossible to complete.
The idea is, what you are being taught is basic riflemanship, and you can learn that with literally any rifle in working condition. The other thing that is taught @ an Appleseed is this country's heritage of riflery, which is being lost, and how important it was in the American Revolution. There is no politicking of any sort, just heritage.
So, yeah, for a 13yo, I'd get an autoloader for Appleseed use. I'd make sure that the stock fits the youngster pretty well, too. Mebbe yr wife can use the same stock. At any rate, bring another rifle for yourself, and do the Appleseed WITH the youngster--it's a great father-and-child thing to do. Enjoy, and please keep us posted!
ETA--As you may know, women and kids shoot for free @ an Appleseed event; likewise military personnel. You WILL need abt. 500 rounds available per shooter for the 2-day event; this partly explains the popularity of the .22 @ Appleseeds. Oh, and elbow pads--did I say elbow pads?