Currently experiencing something similar / same issue.
My Launcher cannot seem to connect to The Sims 3 website.
Does your password contain any special characters? I've read the Sims 3 Launcher cannot recognize special characters in passwords so will not be able to login unless the password is changed to remove them.
Another suggestion I've found online if you're on Windows:
Clear cookies and cache history from your Internet Explorer.
If only Microsoft Edge is available, then delete from there (as Edge has an IE mode)
Click the 3 dots in the top right corner, go to history, and clear history.
Also search for system settings, go to "Privacy, search, and services" and clear your browser histories from there.
This unfortunately did not work for me but seems to have helped other people in the past. Hopefully it's as easy as this for you!