I have a table that keeps track of board meeting votes. I have the following fields:
Resolution ID
Member Name
Vote
I have a table that keeps track of board meeting votes. I have the following fields:
Resolution ID
Member Name
Vote

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If only one member can motion a resolution, you shouldn't have a Motioned? field in this table. It contains a lot of redundant information.
Instead, you should have a field MotionedBy in the Resolutions table. This field should be linked to the MemberID field in the Members table (i.e. the primary key in the Members table.)
This means that Motioned? must also be removed from the subform. Instead, put a combo box on the main form bound to the MotionedBy field, with the Members table as row source.
As a consequence, you don't need to check for multiple members who motion a resolution - the MotionedBy field can contain only one MemberID.
Some remarks on your code:
If rst.NoMatch Then
MsgBox "No County Board Members have been added."
rst.Delete **This Doesn