Maybe get in touch with MTGuns
. They've got used target .22s in stock and have a good rep for supporting the target shooting community. Mac Tilton just retired, though, so the business is being run via Chesebro Rifles
. Either way, contact them, get their feedback, and ask what they've got in their used inventory.
BTW, be aware that in many (most?) cases, formal "target" .22 rifles are single shot. If the type of competition would require a repeater (e.g. CMP Rimfire Sporter), a single shot won't do.
As far as a new rifles, check out Chesebro's CZ452 Scout
: Outfitted with Lyman sights and a forearm rail, it'd make a good starting target rifle for a 10 year old. The CZ452 is an excellent rifle, especially for the price. The Scout version has a shorter barrel & smaller stock. It comes with a single round magazine, but it'll take standard 5- and even 10-shot CZ452 mags. I've never seen it sold with Lyman sights and/or a forearm rail, so this one's specifically set up for youth target shooting, it seems.