i was create my own mobile app. which serve anyone who buy website of booking core , do i need to get any permission for the website author to distribute my mobile app
No permission is required. Customers who purchase templates/themes are free to integrate it with your App. Just don’t brand your item with third party brands of these template/themes, like “App for Theme X”, being X a project of other author. On this specific case, you may need authorization of the third party author.
You can create your own brand and say it is “compatible” with X, Y Z (third party brands), considering you have tested your App with these templates/themes or will offer support the implementation/compatibility process.