Truly, guys use just about anything you can think of at an Appleseed, including bolts and lever-actions, so if you have .22 or centerfire rifle, you're good to go.
That being said, if you're just looking for an excuse to buy another gun, then make sure you get one of the 10/22's without a barrel band up front (such as a Deluxe Sporter, or DSP). Appleseed (which is based on classic high-power techniques) requires a sling to have any chance of shooting the small groups of which you are capable. Fastening the front of the sling to the swivel on the barrel band means your point of impact will shift depending on the exact amount of tension you have on the sling. It's simply a no-win, and by the end of the morning, you will likely have removed the barrel band and be begging for a sling stud to screw into the front of your stock.
If you have time (they used to be back-ordered, mostly due to Appleseed), try to install a Tech Sights TSR100 or TSR200 sight ($70 for the TSR200), which mimics a ghost ring battle sight, and is much more usable than the standard Ruger notch and post (at least for us old guys with old eyes). A DPMS Rapidex windage knob (about $9 from Midway) makes it easier to adjust the Tech Sight windage.
Quick release sling swivels are nice to have so you can detach from the rifle and leave the sling wrapped around your arm while checking targets, but the standard quick release swivels have too big a diameter to fit the standard 10/22 sling studs, so you have to upgrade to larger sling studs to make this work. ($10 for a set of replacement studs and quck-release swivels, according to my records.)
Beyond that, spend as much as you want on hammer, bolt release, and mag release upgrades.
And even further beyond that, don't forget to have fun, and pay attention to the history lessons. For me, the shooting was secondary; hearing the stories of April 19, 1775 were the best part.
––Gene