次のようなコードで自己解決しました。
お騒がせしてすみませんでした。
  Private Sub onBinaryLoad(ByVal sender As Object, ByVal e As OpenReadCompletedEventArgs)
        If Not (e.[Error] IsNot Nothing) OrElse (e.Cancelled) Then
            DirectCast(Me.Controls.Find("TextBox" & e.UserState, True).First, TextBox).Text = New StreamReader(e.Result).ReadToEnd
        End If
    End Sub
    Private Sub Button1_Click(sender As Object, e As EventArgs) Handles Button1.Click
        ServicePointManager.DefaultConnectionLimit = 8
        Dim 変数Max As Integer = 16
        Dim WcList As New List(Of WebClient)
        For i = 1 To 変数Max
            Dim TmpWc = New WebClient
            WcList.Add(TmpWc)
            AddHandler TmpWc.OpenReadCompleted, AddressOf onBinaryLoad
            TmpWc.OpenReadAsync(New Uri("https:XXXX&page=" & i), i)
        Next
        Do
            Application.DoEvents()
        Loop Until WcList.All(Function(n) n.IsBusy = False)
    End Sub