That site does *NOT* use country/state/city drop downs. The city is free form text field, not a drop down.
This is a relatively easy thing to do in JavaScript, but the problem is finding an open source list of all the countries with all their states. You do that, we can help you with the rest.
Whether you do this all in
JS code, without needing AJAX or similar, will depend on the total number of *states* in all the countries on your list. I would say if you have no more than about 1000 states, total, then you can do it 100% on the client with no need for any server-side code, including AJAX.