May a sitting board member who is up for re-election request a copy of the membership list?
This is a matter of state law, but in all probability yes. You don’t say what the director’s purpose is for asking to get the list, but I assume from your question it will be used in some way to promote their candidacy for re-election. I also assume that the corporation has members who vote in that election.
A director generally has a broad right to review all of the books and records of a nonprofit corporation to the extent necessary to perform their function as a director. Even though there may be a personal interest in seeing the information in this case, there are many legitimate reasons why a director would want to know who has the right to vote or who actually does vote in an election. There are also issues relating to the size and characteristics of the membership that are important for the future of the organization and could affect recruiting, membership benefits, and organizational finances.
If I assume that this director is also a member, it is clear that members also have a separate, though usually somewhat more limited, right to the see the books and records of the organization for a proper corporate purpose reasonably related to their interests as a member. Running for office would clearly be a proper corporate purpose and having access to the membership list could be significant in obtaining the votes to get elected.
Nonprofits seldom have contested elections for director in which the candidates actually campaign for the office. It is not considered particularly genteel. But there is no law that prohibits it and it does, in fact, reflect the American democratic process. Elections have consequences.