The vast majority of Appleseed shooters use 22's, but at Ramseur (and many other ranges) any cartridge under 8mm (IIRC) may be fired. It is not unusual for there to be at least a couple of people firing AR's in 5.56, and I've seen more than one dedicated shooter fire 308's or 7.62x54R's or what have you. I don't remember anyone making through a whole day with a mosin--the instructors often have loaner 22's available that start to look pretty attractive to many center fire shooters sometime after lunch on Saturday--but you're welcome to shoot them if you like. Bolt and all. I've never fired an autoloader at an Appleseed, except an AR at a known distance shoot. At the 25 yard events I have only shot my old Marlin 25N bolt gun, and I never felt like it posed a problem for me. There are procedures in place for shooters who bring tube fed guns, too.
It really is a run what you brung program. Instructors will not make fun of you for using a scope!
The only real equipment-based issue is the sling. A lot of the instruction at Appleseed has to do with using the sling as a marksmanship aid, and if you only have a carry strap on your rifle, or worse don't have usable sling studs, you won't be able to take advantage of that knowledge. So if I could make one suggestion for anyone thinking about an Appleseed, it would be to make sure you have a GI-type sling that can be attached appropriately to your rifle.
A cheap foam sleeping bag mat can serve you well for the prone part of the day, FYI. And if you bring elbow pads, don't bring the hard plastic ones, you'll be rolling all around.
If you're chasing the Rifleman patch, then a well-designed Liberty Training Rifle is a help, I will admit. But if you're going there to learn some important lessons about our American heritage and history, as well as to learn some outstanding traditional American rifle marksmanship, then your old Marlin 60 with buckhorn sights will do just fine. Don't stress the gear, just get there.
Not trying to preach, but I wish someone had told me this. It took me a couple of years to put together the money to outfit two 10-22's as reasonable facsimiles of the liberty training rifle so I could let my two older sons shoot them, while I lugged along my old Marlin bolt gun. Silly. Not that the modifications were bad, but they held me back from going out there and getting on the line for a long time. Heck, even if you don't own a rifle at all you can shoot an Appleseed. Contact the shoot boss for the event you'd like to hit and ask about a loaner. There's about a 100% chance you'll be all taken care of.