I'd be really interested in contributing but I'd hate to be contributing and then find out that this non-profit is actually some rich persons tax avoidance scheme or can be pivoted in a way to make money for someone.
Having the source code licensed as GPL or something would make this a smaller concern for me.
Security is security, how you get there is entirely up to your organization. Also I'm not sure how having code access to their site proves that it isn't some rich person's tax avoidance scheme. In a way, all charities allow rich people to avoid paying taxes to some degree.
It is easy enough to have a volunteering agreement that catches this with clauses. Open sourcing the code won’t help with that scenario anyway.
Open source what, exactly? You point makes literally zero sense