Can contacts get my vCard without me having to send it? (Web and Mobile UI)

Yes, they can. 

Your contact can text you the one word "vCard" and MyRepChat will automatically send your vCard to the client as well with this message,

• "Hi {contactname}, Please add my contact information to your phone. If you can't view my virtual contact file, try texting the word VCARDLINK or CONTACTINFO."