This is a very important question for you to remember. If you are contemplating setting up a business website or are looking to redo your current one, make sure this is the first question you ask your prospective marketing or advertising agency. And if they cannot introduce you to their web designer (or design staff) right in their office, you should run.
Most agencies simply are middle men, nothing more. They farm out their web design work to freelancers or other web design companies (while marking up the price for their "expertise"). They're polished salesman. They talk a good game. But do they really know anything about how websites work, how SEO works, how online user habits work...or do they work directly with people who know about web design and development? If not, think twice!
And even worse: What happens if there is something wrong with your site? Are you talking to someone who has no idea what's wrong or how to fix it, who will only pass the information on to someone who doesn't consider your site a priority, or do you want to talk to the person who built your site and can correct the problem quickly to properly serve your customers?
In today's world, your business web presence is absolutely key in any marketing strategy. Newspapers, flyers, billboards, radio, TV...they all take a back seat to YOUR WEBSITE. That's where people go. That's where people are making buying decisions. That's where they learn the real story about who you are and what you do. So are you going to entrust the most important marketing aspect of your business to a company that doesn't have web designers working directly for them? Are you going to have a marketing agency build your website...and they didn't even build their own?