Click on any request of the host bard.google.com and copy the value of Cookie header.
Paste the value in the extension settings vscode-bard.cookies.
OR use shot cut ctrl+shift+B and paste the value.
Note:
Remember to close the opened bard webpage after getting the cookie.
Due to Google's security policy, different regions may require different cookies. You can try setting all cookies starting with __Secure-1, or copy the entire cookie value as in Solution 2.