If a majority of board members are related to members of the union how does the board go about voting to approve the contract?
This situation you describe is not uncommon in Oregon's small school districts. The legislature has recognized a difference in whether a vote would financially benefit a board member or his or her family members or an entire class of people. The legislature has allowed board members to vote on contract approval without stating an actual or potential conflict of interest, because contract approval applies to an entire group (classified or certified staff). (ORS 244.020(11)(b)) Board members should consider how this will be viewed by the public and consider abstaining in some circumstances.
If this was not a contract issue, but an issue involving the approval of hiring an individual closely related to three of five board members, the three would be required to declare an actual conflict of interest and refrain from discussion, deliberation, or voting on the matter. (ORS 244.020(1) and ORS 244.120) As you have noted, there would no longer be a quorum able to approve the hiring. In this situation, the law says that the board members may now declare the conflict, refrain from discussing and deliberating, and vote, so that the decision can be made by the board. (ORS 244.120(2)(b)(B))