No, you don't need a website for your writing business... right away. In fact, you probably don't want to even think about one until you've actually GOT a writing business. See, the thing is, once you get started designing, planning, writing, and paying for your website, it can be a major do.
And chances are, once you fall into that hole, you'll never get out and actually write.
See, most of us are famous for running on the hamster wheel rather than actually being productive. It's a lot easier to look busy than to accomplish something.
Ever heard of paralysis by analysis? Writers tend to live in their heads, and once we're up there, it's like walking through tar.
Once you start working on pulling together a website, it becomes like one of those If You Give a Moose a Muffin books. You'll find a great domain name, but then you'll get stuck deciding on .com, .org, .biz, or every other possible extension. And once you solve that, you'll need a hosting plan. And then a graphic header - which means you'll need a logo. And then you might as well get the logo put onto some letterhead and a business card. But then you'll need a business phone line. And because your cable company's got a deal running, you might decide to get an internet-based phone. And that means you've got to have a installer guy come to your house. But first you've got to clean it. And you've been thinking of using green products, so you'll have to some research first....
See what can happen?
So, the 'no' part of the answer is there to help keep you on track. Go out, get some writing gigs, make some money, get your boat in motion!
Then we'll talk about websites.