If notification credit runs out, do invoices still go out?

They do. The only quota that can run out is bulk announcements — each recipient draws one broadcast credit, and when that balance is empty the send is refused with the reason stated. Transactional messages such as invoices, reminders, and payment confirmations never stop with it.

The Notification Credit and broadcast balances are visible on the school's settings screen along with their usage, and can be topped up without changing the subscription plan.