I use Google Custom Search on my website (JSON API, not generated script tag from Google). On search form submit I send ajax request to https://www.googleapis.com/customsearch/v1?q=QUERY&cx=MY_ENGINE&key=MY_API_KEY
The result is JSON object with all those informations about search results. As it is async, I don't have to refresh page. I just fill
<div id="searchResult></div>
with data I have received.
But how can I handle JSON response on amp page (without refreshing)?