I have been using Capital One which offers virtual credit cards. I have not tried the recommended way by Privacy Action Plan yet so I cannot speak to that. CitiBank also offers the virtual experience (and probably others but am not aware of). What happens is – on line, when you are ready to make a purchase, the CC company generates a virtual card that is linked to your credit card but your actual credit card is not given to the vendor. You can set rules to prevent anyone else from using that number as well as you can schedule it to lock so the vendor cannot charge anything in the future unless you say so.
And, in order to do this, with CapOne, you will download a browser extension, which help to make it easier when shopping.
I hope this helps. It’s my experience.