Applies to: Final Draft 12 / 11 / 10
No, the installers available on our site are for the regular retail version of Final Draft and require a certain type of Customer Number for activation. These numbers are not distributed with App Store purchases.
The version of Final Draft that you bought from the App Store uses a different copy protection system and if you need to get FD on a computer again, you must download and install it from the App Store.
See this article from Apple for details: