The Remington Model 7 in Youth model .243 caliber.
This. If you hit a deer where you should a 243 will kill him as dead as anything. And it will do it without any significant recoil that will affect your kids aim. I despise the "margin of error" argument put forth by people who advocate larger calibers for kids. It's utter horse pucky. A 30 caliber hole through a deers colon will do nothing to increase the odds of recovering that gut shot deer vs a 24 caliber hole through the same area. It's where you hit them that matters waaaay more than what you hit them with. If a kid can't shoot a 243 accurately then I promise you a heavier recoiling caliber will only make that situation worse.
Some one suggested the 257 roberts. A fine caliber but it will do nothing a 243 won't do. In fact the 243 shoots flatter with factory loads. Plus it will not be nearly as easy to find as 243 shells and probably more expensive when you do find it.
Some suggested the 6.5 x 55. Again, a fine cartridge, but of the 4 sporting goods stores within an hour drive of my house maybe 1 would have "a" box of these shells. And that's probably exactly how many boxes they would have, one!
Any store that sells ammo will have 243 in stock and probably multiple types from several makers.