Can I add a picture of the user in a vCard QR Code?

No. A vCard QR Code only allows static text information with a limited number of characters. 

If you want to create a business card with pictures or videos, you can create a QR Code Business Card. All contact data will be stored on the server and not in the QR Code, therefore you have almost unlimited possibilities to store multimedia content.

There is not enough space to include an image in a static QR Code, because a QR Code can contain up to 4096 characters only and an image has way more characters.

You may see on the vCard technical specifications that the photo-field can contain an URL as an alternative and the image will be inserted from the URL. We have tried this option on several devices, however most smartphones do not support this method including Apple iPhones, because of security reasons.