Although the IRS is considerably more interested in governance issues than it used to be, the 501(c)(3) rules do not generally regulate such governance issues.  They are a matter of state law.
If board members are not notified of a meeting and are not permitted to participate in the decisions of the board, those decisions would ordinarily not constitute valid corporate action.  If the board has delegated authority to an Executive Committee or some other committee, the committee could act without notice to the other directors, but it sounds like your situation is much less formal and that some people are actively trying to exclude others.  If the excluded members want to make a fuss about it, they can demand a change in the procedures or, perhaps, even sue to prevent a transaction.
Ultimately, however, it comes down to who has the most votes. If a majority like the course of action approved in the rump phone calls, they can usually ratify or otherwise approve the action at a proper meeting.