That is tricky, since the nature of HTML assumes that the source for an IMG tag is a wholly separate URI, and not inline. I think they might be making allowances for this in XHTML 2.0, but I don't know for sure (and it doesn't exactly help you either way).
As another commenter noted, you could save the content to a file and refer to the file in your SRC attribute. Of course, that will lead to another HTTP request, so you have to either put it in a place that your server already uses for content, have some sort of aliasing set up, or have your script handle both types of request-- the vCard proper and the image attachments. That is what I would do-- have the same script (I assume this is CGI?) that displays the vCard-as-HTML also have a calling-form to fetch the image content. Then return that content with proper Last-Modified, Content-Type, etc. headers and such based on the timestamp of the vCard itself.