There is no mechanism to add an attachment in WebDAV. You will need to create/recreate the item using a WebDAV PUT. The text you would be putting would be the mime of the item, which also contains the attachment. It’s useful here to use CDOSYS to create a message and add an attachment – then extract the MIME from the message and use it for the WEBDAV PUT. You will need a header of "translate" set to "f".
If the message already exists, you will need to, you must do a GET on the message stream, which will give you the message + attachments in a string. Next, modify the stream to include a new attachment. After this, the string can be used in a PUT statement to write the attachment.