Sounds to me like you need to take this over to the forums at
www.RobertsRules.com
. People who know the ins and outs of Roberts Rules of Order will be able to answer more authoritatively than I can, and be able to cite page and paragraph.
As I see it, the question is this: is it necessary for the board to take any action to accept their resignations before they become effective? Or does a failure to refuse the resignation make it effective? And in the absence of any action, how soon do they become effective?
If you have no by-laws, then there can also be the question of authority. Most by-laws will contain a provision that Roberts Rules apply on any matter not otherwise covered. But in the absence of any by-laws, or any specification, do Roberts Rules apply?
I can tell you how it would work in a perfect world...that is, one in which people do things as
I think best. I think that people who quit in a snit and then try to change their mind should learn that they their actions have consequences -- after all, isn't that the exact same lesson we try to teach our kids?