Can I ask my buyer to recommend me to his friends in Chat?
You can ask but I would expect it would be a very unwise thing to do as many will feel pressured, not like that and sooner or later, someone could complain to CS and you find yourself with a Warning.
I suspect you are thinking more about yourself and your want than those of customers.
How would feel if I leaned over you and asked you to tell 10 of your friends about how excellent a fellow I am? Yeah, like that. So why do it to someone else.
If someone wants to help their friend by suggesting you, they just will.