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