You can use the MSXML Base64 encoding functionality as described at www.nonhostile.com/howto-encode-decode-base64-vb6.asp:
Function EncodeBase64(text As String) As String
Dim arrData() As Byte
arrData = StrConv(text, vbFromUnicode)
Dim objXML As MSXML2.DOMDocument
Dim objNode As MSXML2.IXMLDOMElement
Set objXML = New MSXML2.DOMDocument
Set objNode = objXML.createElement("b64")
objNode.dataType = "bin.base64"
objNode.nodeTypedValue = arrData
EncodeBase64 = objNode.Text
Set objNode = Nothing
Set objXML = Nothing
End Function