As a first step, ask Godaddy whether they support AddType directives and if yes, how. That solution would definitely be preferable.
Alternatively, you may be able to work around it using a primitive PHP script that sends the correct headers.
contact.php (untested):
<?php
# Send correct headers
header("Content-type: text/x-vcard");
// Alternatively: application/octet-stream
// Depending on the desired browser behaviour
// Be sure to test thoroughly cross-browser
header("Content-Disposition: attachment; filename=\"contact.vcf\";");
# Output file contents
echo file_get_contents("contact.vcf");
?>
this would serve the VCF file with the correct header.
