I agree,10-22 with the short barrel and a collapsable [AR TYPE] stock. I set one up for
my petite little 8 year old granddaughter and she has a 10" length of pull. The Blackhawk
stock has a 10.5" LOP collapsed, so she can shoot it now and both can grow together.
Blackhawk calls theirs the Axiom.
I was sort of thinking along the same lines. You'll be hard pressed to truly find a semi-auto with a small profile and weight like the crickett. If it were me, and I had to have what you are seeking then I would get the Interarms/Norinco clone of he above Browning rifle and cut it down. The Interarms clone is cheap and has little to no collector value, thus you could shorten the barrel to 16" if desired and shorten the LOP. It would be a pretty cool little rifle!
Here is what it could look like cut down a little.
I'm a fan of going the 10-22 route. They have something they can shoot now, will grow with them, and once they get full size there is a multitude of available parts to customize to their liking.
Another nice feature of the 10/22 is the availability if carbon fiber barrels. I set one of these up for my, 7 year old, daughter. The reduced weight of the carbon fiber barrel really works well for her.
Kinda surprised that no one has mentioned the little Ruger 10/22 Compact as a candidate. A bit larger than a Chipmunk, Cricket, et alia but still quite significantly shorter and lighter than the standard carbines. Better set of open irons on it, too: FO Williams front and rear.
I traded for one when my younger nephew was still too small to comfortably handle any of my "adult" sized .22 RF rifles, including a Rossi M62A with the 'youth' butt stock. Worked out very well, and allowed him to compete with his older brother on the duelling tree targets on even terms.