A foundation may only incur expenses for meetings/events that are related to carrying out its exempt purposes (including reasonable and necessary administrative expenses). If a board member wants to invite all board and family members to a family event, the foundation may only reimburse the board member for the expenses incurred if the event is for the purposes of the foundation.
The burden is on the foundation to prove this. As an initial matter, and without knowing any facts, it would seem that an event at which family members who are not involved in the management of the foundation are present would not qualify as an event that is for the purposes of the foundation. In that case, it would be unlikely that the foundation could reimburse the expenses incurred by the board member.