evo ti rešenje, nadam se da će ti malo proširiti pogled na programiranje
Private Sub Button2_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button2.Click
        Dim duzStr, br1 As Integer
        Dim strMidString As String
        br1 = 0
        duzStr = Len(TextBox1.Text)
        For i = 1 To duzStr
            strMidString = Mid(TextBox1.Text, i, 1)
            If strMidString = "a" Or strMidString = "e" Then
                br1 = br1 + 1
            ElseIf strMidString = "i" Or strMidString = "o" Then
                br1 = br1 + 1
            ElseIf strMidString = "u" Then
                br1 = br1 + 1
            End If
        Next
        TextBox2.Text = br1
    End Sub
pozz!