As Mark C points out, you can use the MSXML Base64 encoding functionality as described here.
I prefer late binding because it's easier to deploy, so here's the same function that will work without any VBA references:
Function EncodeBase64(text As String) As String
Dim arrData() As Byte
arrData = StrConv(text, vbFromUnicode)
Dim objXML As Variant
Dim objNode As Variant
Set objXML = CreateObject("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