Can I be on the board of directors for a nonprofit while on active duty? What regulation would cover this?

In a paid or non-paid capacity? Depending on the role and compensation it changes how things are viewed, and what "potential" conflicts of interest (and regulations) you may have.

If you are not being compensated, then you are essentially doing Volunteer Work.

If you are being compensated, then coordinate with the command.

AR 210-22 covers NPOs on Army Installations, but that only applies to on-post. You should have a NPO coordinator on post who can help you determine what NPOs you can join and in what capacity.

1LT (Join to see) , Your Office of the Staff Judge Advocate will have an Ethics Counsel on staff who can address your question specifically. There are limitations in place in the Joint Ethics Regulation as to service in Non-Federal Entities. You should discuss your particulars with an Ethics attorney.