Since they don't support url queries you'll have to reproduce it with curl, but

    curl 'https://tuxdex.com/' \
      --data-raw 'q=new+anonymous+search+engine+domain%3Anews.ycombinator.com&ilang=en&lang=en' \
      -H 'User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:103.0) Gecko/20100101 Firefox/103.0'

cheerfully returns "Unfortunately we have no content for your request" and nukes my text input so I have to start typing from scratch