When I’m out glove hunting at yard sales and thrift stores I see youth catchers mitts fairly often- some in good shape, others not so good. My suggestion would be, for a kid that age, just getting going at the position, try to find a few second hand mitts to try, without spending a lot of money-they’re out there.
I would imagine you could find a handful that are in decent shape and already broken in for him to use as he learns the position, so you don’t spend a lot on a brand new one that needs breaking in. With the summer winding down, the next several weeks and months might be good glove hunting at second hand stores.
There are also good youth gloves out there real cheap simply because they need a little lace work done, which are always fun to find, fix, and see get used around the neighborhood. I tell people where I find a lot of my gloves, and some tell me they would never set foot in a thrift store or be caught dead at a yard sale.

Give us the gate key. I have no gate key.
Fezzik, tear his arms off. Oh, you mean this gate key!