What is the best way to transfer authorization on the site (cookies) to another computer and put it on the pipeline?

Question

HTTP Cookies

There is a third-party site, you can log in to it only through the social. networks, no mail and passwords are used.
Task: organize the distribution of authorized accounts of this site so that the user can enter it through his computer / browser.
When manually transferring cookies, everything works, it is not tied to ip.
What is the best way to do all this so that it is with a minimum of hassle for the consumer?
PS Is it possible to build it all through your anonymizer (php or c#)?