I have the same issue with Thunderbird. Exported my contacts on my (old) phone to file 00002.vcf. Send the file with GMail on the phone to my email account (on yahoo IMAP). Much to my surprise the file did not show as attachment in Thunderbird. Checked on GMail (in browser from pc) and saw in send folder that the attachment was in fact there.

Then back to Thunderbird and in the received mail selected "Show sourcecode" and can see that the attachment in fact is in the mail but does not show in Thunderbird (GUI): See image

The problem might be due to the content-type 'application/octet stream':

http://kb.mozillazine.org/Actions_for_attachment_file_types#application.2Foctet-stream

I suggest you open Config. editor from Options/Advanced/General, and double-click the preference mailnews.display.show_all_body_parts_menu to true. Then, select All Body Parts under View/Message Body As and see if the vcf attachment is shown (possibly with a txt extension).