Write text file in appending (utf-8 encoded) in VB6

You could combine binary I/O with an API call to perform the conversion to UTF-8:

Option Explicit

Private Const CP_UTF8 As Long = 65001

Private Declare Function WideCharToMultiByte Lib "kernel32" ( _
    ByVal CodePage As Long, _
    ByVal dwFlags As Long, _
    ByVal lpWideCharStr As Long, _
    ByVal cchWideChar As Long, _
    ByVal lpMultiByteStr As Long, _
    ByVal cchMultiByte As Long, _
    ByVal lpDefaultChar As Long, _
    ByVal lpUsedDefaultChar As Long) As Long

Private Function OpenAppendUTF8(ByVal FileName As String) As Integer
    OpenAppendUTF8 = FreeFile(0)
    Open FileName For Binary Access Write As #OpenAppendUTF8
    Seek #OpenAppendUTF8, LOF(OpenAppendUTF8) + 1
End Function

Private Sub WriteUTF8( _
    ByVal FNum As Integer, _
    ByVal Text As String, _
    Optional ByVal NL As Boolean)

    Dim lngResult As Long
    Dim UTF8() As Byte

    If NL Then Text = Text & vbNewLine
    lngResult = WideCharToMultiByte(CP_UTF8, 0, StrPtr(Text), Len(Text), _
                                    0, 0, 0, 0)
    If lngResult > 0 Then
        ReDim UTF8(lngResult - 1)
        WideCharToMultiByte CP_UTF8, 0, StrPtr(Text), Len(Text), _
                            VarPtr(UTF8(0)), lngResult, 0, 0
        Put #FNum, , UTF8
    End If
End Sub

Private Sub Main()
    Dim F As Integer

    F = OpenAppendUTF8("test.txt")
    WriteUTF8 F, "Hello"
    WriteUTF8 F, ChrW$(&H2026&)
    WriteUTF8 F, "World", True
    Close #F
    MsgBox "Done"
End Sub