Welcome to the Make community!
You’ll have to set up a intermediary server outside of Make as a go-between so that that user can click “sign in”, then the server passes the completed authorization and refresh tokens to Make, and then your scenario will store their OAuth keys somewhere like a Data Store.
Did you know, this forum has a Hire a Pro category, where you can post your request for off-site specialised help on other platforms (video call/screenshare/private messaging/etc.)? This may help you get your issue resolved faster especially if it is urgent. It is important to post your request in the Hire a Pro category, as forum members are not allowed to advertise their services elsewhere (like here).
Hope this helps! Let me know if there are any further questions or issues.
— @samliew
P.S.: Did you know, the concepts of about 70% of questions asked on this forum are already covered in the Make Academy. Investing some effort into it will save you lots of time and frustration using Make later!