I did my first Appleseed to check it out as a possible activity for my children. I never expected to find it so rewarding, personally

In any event - it was obviously a great opportunity for my kids. Mary was 6 at the time. I picked her up a little stainless Cricket. We went through a few iterations before we found the perfect setup ...
Started with a rolled up shooting mat. That worked OK.
Tried sandbags - not quite tall enough.
Put a bipod on the rifle - found that she tends to push the rifle, which levers the stud out of the stock pretty quickly.
We ultimately settled on the perfect Cricket setup: one of those clamp on rail things from Amazon (cheap, chinese aluminum, < $10) like one would use to mall-ninja up a 10/22 (as one does), and a likewise cheap chinese-made picatinny mount bipod, also from Amazon.
That, my friend, did the trick. Rock solid, stable, and can take a lot of abuse. I keep several of the bipods and clamp on rails in my oh-crap-kit for Appleseeds now, in case I run into another kid with the same problem. One important thing, though - you have to be careful to get a very low profile one, meaning no top rail ... I had to try two or three, before I found one low enough to not obstruct the Cricket's front sight.
Don't disparage the lowly Cricket - after 3 or 4 Appleseeds, Mary was completing Stage 1 and Stage 4, and completing Stage 2 and 3 at least 50% of the time - meaning sending all of her rounds downrange, aimed. Watching her run that little single shot so fast (tiny fingers help, I'm sure) is one of the coolest things I've ever seen. She is, of course, the designated marksman ... must remain prone. But it's a great little rifle.
Here's a thought - for the kids to have a good time, it's important for them to feel like they've achieved something. A six year old, in their first Appleseed or two, is unlikely to shoot impressive scores (the first couple we were celebrating when she was on paper), especially with a Cricket (so-so sights and a short sight radius ... I can't shoot particularly respectable groups with it, myself). So learning to feed the single shot quickly gave her something to be proud of ... it was low-hanging fruit, an achievement, that kept her from getting discouraged and quitting. Folks would often comment on it when they saw it, which gave her confidence and made her feel proud, which helped motivate her to do Appleseed after Appleseed.
At the last Appleseed, she was shooting around a 170 or so ... she'll hit 210 at her next one, I'm sure ... I've been thinking of having some (obviously unofficial) "Designated Marksman" patches to give her when she does - I'm thinking just the silhouette of one of those prone green army men with the words "Designated Marksman" under it - she really covets my Rifleman patches, and wants one of her own badly ...