Obviously, the more talented you are, the more opportunities will pass your way. I consider myself to be a decent writer. Recently, I have taken to mastering Web construction in an attempt to increase the number of skills that I have. Can you construct Web sites? How important a skill do you think it is for a copywriter's success?
It's not that hard to install Dreamweaver and Front page... then review the tutorials to learn how to put a simple web page together. You can also get EditPlus - very cheap option for web designing software, and can get templates for sales copy off the web to load up. It's not too hard and can make a BIG difference!
I think anyone providing web content, be it words or graphics, should have at least a basic understanding of web programming. You should generally know what sorts of things can be fixed with a quick edit vs those things that require significant programming. A writer should understand what meta tags, alt text and heading tags are and how they impact SEO, usability and the overall effectiveness of the page. Does a copywriter need to know how to program a site from the ground up? It doesn't hurt. And besides, don't you need some skills to maintain your own website?