I've been part of other not-for-profit orgs with countercultural missions. It's always been possible to get a few members -- not on the board -- to step forward and do a yearly audit of the books. These audits aren't high-end finance work products: you'd have to pay big bucks to somebody like PWC for that. But they do spot stuff like misuse of payment cards.

It's interesting how members who have finance experience, and members who want to learn something about finance, both step forward. A good volunteer audit can be part of the educational mission of the org.