I'm updating the database in the ItemCommand, because the updating should happen when the LinkButtons are clicked.

Protected Sub ListView1_ItemCommand(ByVal sender As Object, ByVal e As System.Web.UI.WebControls.ListViewCommandEventArgs) Handles ListView1.ItemCommand

        If e.CommandName = "Agree" Then
            Using conn As New SqlConnection(ConfigurationManager.ConnectionStrings("ConnectionString").ConnectionString)
                Using comm As New SqlCommand("UPDATE Table1 SET Agrees=Agrees+1 WHERE ID=@ID", conn)
                    comm.Parameters.AddWithValue("@ID", e.CommandArgument)
                    conn.Open()
                    comm.ExecuteNonQuery()
                End Using
            End Using
        End If

        If e.CommandName = "Disagree" Then
            Using conn As New SqlConnection(ConfigurationManager.ConnectionStrings("ConnectionString").ConnectionString)
                Using comm As New SqlCommand("UPDATE Table1 SET Disagrees=Disagrees+1 WHERE ID=@ID", conn)
                    comm.Parameters.AddWithValue("@ID", e.CommandArgument)
                    conn.Open()
                    comm.ExecuteNonQuery()
                End Using
            End Using
        End If

    End Sub

The LinkButtons are disabled in their OnClick events.

Protected Sub LinkButton_Click(sender As Object, e As EventArgs)
        Dim lb As LinkButton = DirectCast(sender, LinkButton)
        If lb.ID = "Agree" Then
            Dim lb2 As LinkButton = DirectCast(DirectCast(lb.NamingContainer, ListViewItem).FindControl("Disagree"), LinkButton)
            lb.Enabled = False
            lb2.Enabled = False
        Else
            Dim lb2 As LinkButton = DirectCast(DirectCast(lb.NamingContainer, ListViewItem).FindControl("Agree"), LinkButton)
            lb.Enabled = False
            lb2.Enabled = False
        End If
    End Sub