This is rather complicated, can we write (a field and buttons and all that) a search engine inside a flash movie so that people can search the site or the web? I guess I am asking for a flash version of JavaScript or php search engine.

Yes you can. It isn't all that much work really. Just need to make an interface, and set your buttons. You can use JS and php to read/write to/from Flash and show the contents there.

Basically you can set it up with all JS and add the Flash interface ontop of it.