This seems similar to what you’re talking about. The only thing it doesn’t have is the ability to reward with bitcoin. It does allow you to charge an account and make purchases directly to Amazon. It's an online tool but if you could convince them to accept bitcoins they could interface directly with spendbitcoins.com to buy Amazon gift cards and use them to purchase the rewards. Maybe Steve from Bitpay could call them to discuss the possibility and help them set it up if they’re interested.
There are many parents on this forum and it would be a good project for Bitcoin. If myjobchart.com was convinced to accept BTC it would be good for them as they would have a community of customers ready to immediately spend for the purchase of future rewards for their kids. Paying children in BTC would also build a future generation ready to accept transfer of value using BTC as second nature (like spending fiat).
-seriously, though- I don't think they'll accept BTC. It looks like their revenue stream relies on getting kids to open accounts @ ING. However, they also snag some referral fees from Amazon which, as claimed by a random CSR agent, will be accepting Bitcoin "soon." I have a feeling ING paid a LOT for integration, however. That said, given kids are assigned a "points" balance, there's no reason "points" couldn't be tallied off-record and made to represent Satoshis. When kids request cash-outs, parents are effectively just sent a notice that the kid wants money and it's time to pay up -- really no reason the parent simply couldn't assume "points" = Satoshis (though this makes the "saving" option via the site impossibru).
Otherwise, it doesn't allow me to assign priority of task ("Jobs" and "Extra Jobs" categories are acceptable, though), assign due/expiration dates, assign multiple-worker tasks, assign customized user groups, allow workers to note they're working on a task so others don't waste their time preparing, allow the admin/manager to dispute the quality of work (or even that it was done), there is no way to deduct points as punishment, and work is not pushed to workers (outside of just appearing on the screen, which I guess they're supposed to always have open). I'm still leaning toward a bug-tracker.