I am not aware of any federal or state law that would prohibit a nonprofit employee from campaigning for candidates for election to the organization’s board of directors.  The federal tax restriction on 501(c)(3) electioneering applies to organizational involvement in elections for public office, not personal involvement in internal organizational elections.  (See Ready Reference Page: “IRS Issues New Guidance on Electioneering”) I have never heard of any state law that would prohibit it and am not familiar with membership organizations that have bylaws or personnel policies that prohibit it.

Whether it is a wise idea is a totally different question.  For the same reason that governmental employees are usually prohibited from participating in partisan political activity, it could lead to a lot of abuse if employees of a charity were active — or could be forced to be active — in the internal elections of directors. Obviously, employees who are members should have the right to vote, like civil servants of governments. But if the organization needs a professional staff to carry out its mission, publicly taking sides in contested elections is not likely to promote their professionalism.