No, you don't need to have your own website to use Donorbox. 

Every Donorbox donation form you create gets its own stylish donation page, hosted on our secure SSL-enabled servers. It is completely free and you can have as many as you need. 

A dedicated URL will be allocated to your donation page, which you can also edit, if needed. And it will always be available to you until you decide to archive your campaign. When a donor clicks your campaign's URL, they'll be taken directly to your Donorbox-hosted donation page. 

You can read more about Donorbox donation pages here.