Reddit Secret Santa has been going on since 2009 and it allows you to do just that!
Essentially you sign up to the website, get matched up with a random person, send them a gift and in return you receive a gift from a different random person.
I have participated in 16 of these events since 2009, check out my profile here.
How the hell have I done 16 you ask?
Well throughout the year they do many other events as well, for instance there are Halloween, book, comic book, craft and even tshirt exchanges, though the Secret Santa ones tend to draw the biggest crowds.
Last year RedditGifts had the BIGGEST online secret santa exchange, at 30,250 people.
Sign up here (This is a referral link to me)
Click HERE if you want to sign up, but dont want to link to me
Feel free to ask me any questions.
Here is there official FAQ.
However, right off the bat, let me say that I have NEVER heard of anything creepy or illegal happening to anyone who has signed up.
So you have no reason to worry about identity.
Also this event is open to ALL countries, I don't think there are any exceptions.