There are a few manufacturers that build youth rifles or models that come with the stock spacers that you can add as the youth's LOP changes so that fit remains proper as they grow. This will make a difference in their comfort and felt recoil levels and allow them to shoot the rifle better over the long run. Let him try a few different models to see which fits him best.
As for cartridges, there are a number of good cartridges out there that will work for moose. They are not overly tough or tenacious of life...but a mud caked bull in the rut after wallowing can make it tougher on a bullet to penetrate (as John Nosler found out and then designed the Partition bullet for controlled expansion and penetration). This is why they recommend a retained energy rating of 1500 ft. lbs of energy at the target for moose. And in the event that the bullet encounters heavy bone, there is enough energy to break bone and still get to the vitals. (Of course, there is a substantial difference in moose body size dependent upon the subspecies (Shiras, Canadian, or Alaska/Yukon) you'll be hunting in your local area.)
The 6.5s (6.5x55, 6.5 Creedmoor, 260 Rem) would be a good starting point with 130-140 gr bullets (AccuBond, Terminal Ascent, etc.) out to about 300-375 yards, dependent upon bullet weight.
The 7mm-08 Rem (or 7x57/275 Rigby) is also a good cartridge, and the 140 gr bullets a good starting point, but you can go up to the 154 gr bullets too (up to 175 gr in the 7x57), out to about 375 yards, again dependent upon bullet weight.
The 308 Win will also work, either with a premium 165 gr bullet, or the tried and true 180gr bullets, out to 400 yards or so, again dependent upon bullet weight.
If you can arrange it, let him shoot some different rifles at the range on the metal gongs (if they have them at 100 and/or 200 yards) to see which he prefers. And have him try different shooting positions, to mimic field positions, not just from the bench.
Did this with my daughter (who is 5' 4") and in the end, she chose a Rem 700 Titanium in 270 Win; after shooting a 260 Rem, a 7mm-08, a 270 Win, a 308 Win, a 338 Federal, and a 358 Win in a variety of rifles. (She took her first deer with my BLR in 358 before this, as we were out hunting and it was what I had with me that day, and had tried it out to see how she managed it). She can consistently produce 1" groups at 100 yards with it, and has since taken moose and stones sheep with it.