In order to download iMovie you would need to log into the app store with the Apple I.D. that you have been using. Are you saying that you can open the app store but it will not recognize your Apple I.D? Can you describe what happens when you try to log in? Any error message? Have you tried clicking on Apple Logo/App store? Perhaps you are already logged in.
Assuming that you can log in, here is a procedure that should work to download the current version of iMovie: To update/redownload iMovie 10, drag your iMovie 10 app from the Applications folder into the trash, but do not empty the
trash. Sign in to the app store and open your purchases folder by clicking on your name at the bottom of the app store side bar. You will see your purchases displayed, including iMovie 10. Redownload/update iMovie from there. If not successful, drag your present iMovie app out of the trash and return it to the Applications folder.
-- Rich