@Kathrine3 Hi Kathy - fellow Vanouver host here. No, as far as I'm aware, there is no permit or registration requirements for operating in Vancouver or Clark County. Airbnb currently collects and remits on our behalf the following taxes:
General Sales and Use Tax (Washington)
General Sales and Use Tax (Clark Cty)
Accommodations Tax (Clark Cty)
Local Sales and Use Tax (Public Transportation Benefit Area)
I wouldn't be surprised if additional registration and permit requirements make their way across the Columbia, though, as Portland has a lengthy process. Let me know if you have other questions!