Does this get you started:

Sub Test()
Dim i As Integer
Dim strList As String

For i = 1 To Range("MyRange").Count
strList = strList & """" & Range("MyRange")(i).Text & """, "

Next i
strList = Left(strList, Len(strList) - 2)
MsgBox strList
End Sub

I tested this on your short list and it produced the correct string.
However there may be a limit to the length of the string (max number of
characters)
Don't know if this will work for you.

" wrote:

Is there a way to select a column of ID numbers in an Excel (2002)
column and save it to a text file or clipboard with comma quote
delimiters?

0000031814
0000044857
0000015098
0000043186
0000111012
0000017184

becomes

"0000031814","0000044857","0000015098",
"0000043186","0000111012","0000017184"