Who needs to get an ISBN for the book you have written? The ISBN is part of the publishing process. The ISBN is there to uniquely identify a particular edition of your book through the book trade. As part of the publishing process, the ISBN is the responsibility of the publisher of your book. But perhaps you say that you are not having a publisher, that you're doing it all yourself. In that case, you will take off your writer's hat and need to find one for yourself as publisher.
When you apply to your region's ISBN agency to purchase ISBN numbers, you apply as the publisher of a work. You may well be the author of the work and are going to do everything else as well. But it is as the publisher of the book that you obtain ISBNs. Your ISBN agency is well used to new publishers and writers that are getting into the publishing business as self-publishers of their own work.
The International ISBN Agency appoints national and regional agencies to work with publishers throughout the world. There is only one ISBN agency for each national or regional jurisdiction. You can find a complete list of them on our ISBN Agency page. It's a long page, so use your browser's search facility (usually Ctrl+F) to find your locale.