You should follow the corporation's by-laws. The resigning board members should ideally do so in writing. Then you should have resolutions appointing (or electing) the new board members. Your conflict of interest policy should also be adopted by resolutions, if not done already. You do not need to report the change to the state or the IRS, but double check with the Attorney General's office about your filing requirements as they have oversight over not-for-profits (most usually with respect to your fundraising activities).
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We agree with Attorney Kurzon that the starting place is your by-laws which set out procedures for appointing new board members. You may find that your by-laws require the organization to have a specific number of directors or a range is set out (5-9 directors). Many not-for-profit organizations require a potential board member to complete an application listing qualifications for service and prior or concurrent boards on which the applicant has served. Obviously, these two directors have already accepted the offer to serve, but it would be good to establish the application as a practice in the future. As noted, written resignations from the two departing directors are preferred, but at a minimum, the fact that they have resigned should be referenced in minutes of the entity at the next board meeting. Depending on the size (financial) of your agency, the directors (and potentially the change of directors) may be required to be disclosed on the Form 990 that is filed by the organization.
My two colleagues have given you very good advice: 1) Bylaws 2) Minutes 3) filings with the state to indicate current directors. Try to find a lawyer to sit on your Board who has a passion for your charitable purpose to benefit from ongong matters such as the above.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
Everyone's answer is correct: follow your bylaws, record decisions in your minutes. If you only operate in NY, your 990 and NY CHAR 500 (the NY annual filing) will list your board otherwise no other action is needed with the State or Feds.