I just tested this on a fairly old iPad (iPad 2 or 3 ???) with 9.3.1 installed and it worked fine. So it seems to work fine. Honestly I am surprise a little because iOS did not use to support vCards at all unless you emailed it to yourself. I like that improvement to iOS but if it is not working for you, unfortunately, there is nothing I can do to fix the issue on your iPhone.
My apologies for not having a solution for you!