As far as I know there are two types of notebooks that are presently available at the Try page at Jupyter.org. (I assume you meant that?):
One is the current one that runs in JupyterLite that would be running inside your browser using the Pyodide kernel. I’m not sure the WASM version works with that extension; however, it also has limited abilities and I suspect the Jupyter ai is for actual Python kernel at this time and not the more limited Pyodide kernel. I could be wrong.
The other option are sessions in your browser available via the MyBinder service using modified JupyterHub. That runs on a remote virtual computer without the need for login. Since it is remote and there is no secure login, there is a potential very slim chance your session could be monitored/hijacked and so you should not be sharing keys in order to use Jupyter AI there. See here about ‘Securitu and Privacy’.