1) Parents are charged via typical merchant mechanism to fill their account with "credits" worth $0.99 each, costing $1.00, and can optionally allow the app to automatically top-up their account when low.
2) The service sends cash in a registered mail envelope at a frequency configurable by users, with mailing fees paid for by credits.
Then the business can function and grow a user-base, and by holding the balance, the company can build up a lot of cash-on-hand which will enable them move onto the bank integration they originally wanted. Bonus: receiving legitimate mail and getting cash in hand are very exciting experiences for a kid.