There is certainly no federal tax reason that you cannot open your board meetings with prayer or provide religious guidance to your clientele. Religious organizations of all sorts have regularly been recognized as 501(c)(3) charitable organizations.

If you are receiving food that you distribute under some kind of governmental contract, it could be impermissible under the contract to dispense religious ideas along with the food. There are significant Constitutional issues with government support of religious activity.  But even those limitations are unlikely to prohibit prayer at the start of board meetings so long as it is an internal meeting and is not forced on the consumers of the service. Ask the executive director for a citation of authority for her statements.