Dynamic Taxonomy Filter dropdown with childs in Drupal 7
You need to use Ajax and the drupal form api to build this type of search form.

I recommend looking at the Taxonomy Edge module to make "finding the children terms of some parent" really easy, because the SQL can be annoying. Then you can just focus on the Ajax needed. Drupal has great support for Ajax!

I recommend looking here on drupal.stackexchange to find ajax examples of which there are plenty. I'm nearly positive there is an example of this type of ajax "update the form with some new options or div" in the Drupal 7 Developer Examples module.