Taking a quick look at the network traffic, the site does a POST to https://america.gov/api/chat with a JSON payload when submitting a query:

  {"language":"en","id":"VSvMeyCMKTX05oLW","messages":[{"parts":[{"type":"text","text":"how do I register my new business?"}],"id":"oHub5UDJ3iYARhao","role":"user"}],"trigger":"submit-message"} 

So the user's query is hitting a server so to answer my own question, the answer is No to local inference / models.

Edit: PDF attachments and voice input are processed locally according to the site's privacy policy[1]. But they still use 3rd party 'AI Providers' that abide by a ZDR.

[1] https://america.gov/privacy-policy