I had a similar issue when I am trying to log in VS code leetcode extension. Then I used another way of leetcode login using VS code i.e. Leetcode Cookie. In this method, you need to use your leetcode username and then it will ask for the browser cookie of the leetcode website.

To copy the cookie you need to go to the browser network tab and grab the first call of Graphql endpoint and copy the whole content of the cookie & paste this into VS code login where it's asking for the cookie to enter. I have attached the screenshot below for the cookie info.

Once you have given the all info it will list all the problems available in leetcode in VS code editor.